Class 12th Mathematics Part Ii CBSE Solution
Exercise 13.1- Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) =…
- Compute P(A|B), if P(B) = 0.5 and P (A ∩ B) = 0.32
- If P (A) = 0.8, P (B) = 0.5 and P (B|A) = 0.4, find (i) P(A ∩ B) (ii) P(A|B)…
- Evaluate P(A ∪ B), if 2P(A) = P(B) = 5/13 and P(A|B) = 2/5.
- If P(A) = 6/11 , P(B) = 5/11 and P(A ∪ B) = 7/11, find (i) P(A∩B) (ii) P(A|B)…
- A coin is tossed three times, where (i) E : head on third toss, F : heads on…
- Two coins are tossed once, where (i) E : tail appears on one coin, F : one coin…
- A die is thrown three times, E : 4 appears on the third toss, F : 6 and 5…
- Mother, father and son line up at random for a family picture E : son on one…
- A black and a red dice are rolled. (a) Find the conditional probability of…
- A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5}…
- Assume that each born child is equally likely to be a boy or a girl. If a…
- An instructor has a question bank consisting of 300 easy True / False…
- Given that the two numbers appearing on throwing two dice are different. Find…
- Consider the experiment of throwing a die, if a multiple of 3 comes up, throw…
- If P (A) = 1/2, P(B) = 0, then P (A|B) isA. 0 B. 1/2 C. not defined D. 1…
- If A and B are events such that P(A|B) = P(B|A), thenA. A ⊂ B but A ≠ B B. A =…
Exercise 13.2- If P(A) = 3/5 and P (B) = 1/5, find P (A ∩ B) if A and B are independent…
- Two cards are drawn at random and without replacement from a pack of 52 playing…
- A box of oranges is inspected by examining three randomly selected oranges…
- A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on…
- A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event,…
- Let E and F be events with P (E)= 3/5, P (F) = 3/10 and P (E ∩ F) = 1/5. Are E…
- Given that the events A and B are such that P(A) = 1/2, P (A ∪ B) = 3/5 and…
- Let A and B be independent events with P (A) = 0.3 and P(B) = 0.4. Find (i) P(A…
- If A and B are two events such that P (A) = 1/4 , P (B) = 1/2 and P (A ∩ B) =…
- Events A and B are such that P (A) = 1/2, P(B) = 7/12 and P(not A or not B) =…
- Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6. Find…
- A die is tossed thrice. Find the probability of getting an odd number at least…
- Two balls are drawn at random with replacement from a box containing 10 black…
- Probability of solving specific problem independently by A and B are 1/2 and…
- One card is drawn at random from a well shuffled deck of 52 cards. In which of…
- In a hostel, 60% of the students read Hindi newspaper, 40% read English…
- The probability of obtaining an even prime number on each die, when a pair of…
- Two events A and B will be independent, if(A) A and B are mutually exclusive…
Exercise 13.3- An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour…
- A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black…
- Of the students in a college, it is known that 60% reside in hostel and 40% are…
- In answering a question on a multiple choice test, a student either knows the…
- A laboratory blood test is 99% effective in detecting a certain disease when it…
- There are three coins. One is a two headed coin (having head on both faces),…
- An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000…
- A factory has two machines A and B. Past record shows that machine A produced…
- Two groups are competing for the position on the Board of directors of a…
- Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three…
- A manufacturer has three machine operators A, B and C. The first operator A…
- A card from a pack of 52 cards is lost. From the remaining cards of the pack,…
- Probability that A speaks truth is 4/5. A coin is tossed. A reports that a…
- If A and B are two events such that A ⊂ B and P(B) ≠ 0, then which of the…
Exercise 13.4- X 0 1 2 P(X) 0.4 0.4 0.2 State which of the following are not the probability…
- X 0 1 2 3 4 P(X) 0.4 0.5 0.2 -0.1 0.3 State which of the following are not the…
- Y -1 0 1 P(Y) 0.6 0.1 0.2 State which of the following are not the probability…
- Z 3 2 1 0 -1 P(Z) 0.3 0.2 0.4 0.1 0.05 State which of the following are not…
- An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X…
- Let X represent the difference between the number of heads and the number of…
- number of heads in two tosses of a coin. Find the probability distribution of…
- number of tails in the simultaneous tosses of three coins. Find the…
- number of heads in four tosses of a coin. Find the probability distribution of…
- Find the probability distribution of the number of successes in two tosses of a…
- From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn…
- A coin is biased so that the head is 3 times as likely to occur as tail. If the…
- A random variable X has the following probability distribution: X 0 1 2 3 4 5 6…
- The random variable X has a probability distribution P(X) of the following…
- Find the mean number of heads in three tosses of a fair coin.
- Two dice are thrown simultaneously. If X denotes the number of sixes, find the…
- Two numbers are selected at random (without replacement) from the first six…
- Let X denote the sum of the numbers obtained when two fair dice are rolled.…
- A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18,…
- In a meeting, 70% of the members favour and 30% oppose a certain proposal. A…
- The mean of the numbers obtained on throwing a die having written 1 on three…
- Suppose that two cards are drawn at random from a deck of cards. Let X be the…
Exercise 13.5- A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the…
- A pair of dice is thrown 4 times. If getting a doublet is considered a success,…
- There are 5% defective items in a large bulk of items. What is the probability…
- Five cards are drawn successively with replacement from a well-shuffled deck of…
- The probability that a bulb produced by a factory will fuse after 150 days of…
- A bag consists of 10 balls each marked with one of the digits 0 to 9. If four…
- In an examination, 20 questions of true-false type are asked. Suppose a student…
- Suppose X has a binomial distribution b (6 , 1/2) . Show that X = 3 is the most…
- On a multiple choice examination with three possible answers for each of the…
- A person buys a lottery ticket in 50 lotteries, in each of which his chance of…
- Find the probability of getting 5 exactly twice in 7 throws of a die.…
- Find the probability of throwing at most 2 sixes in 6 throws of a single die.…
- It is known that 10% of certain articles manufactured are defective. What is…
- In a box containing 100 bulbs, 10 are defective. The probability that out of a…
- The probability that a student is not a swimmer is 1/5. Then the probability…
Miscellaneous Exercise- A and B are two events such that P (A) ≠ 0. Find P(B|A), if: (i) A is a subset…
- A couple has two children, (i) Find the probability that both children are…
- Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person…
- Suppose that 90% of people are right-handed. What is the probability that at…
- An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15…
- In a hurdle race, a player has to cross 10 hurdles. The probability that he…
- A die is thrown again and again until three sixes are obtained. Find the…
- If a leap year is selected at random, what is the chance that it will contain…
- An experiment succeeds twice as often as it fails. Find the probability that in…
- How many times must a man toss a fair coin so that the probability of having…
- In a game, a man wins a rupee for a six and loses a rupee for any other number…
- Suppose we have four boxes A, B, C and D containing coloured marbles as given…
- Assume that the chances of a patient having a heart attack are 40%. It is also…
- If each element of a second order determinant is either zero or one, what is…
- An electronic assembly consists of two subsystems, say, A and B. From previous…
- Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black…
- If A and B are two events such that P(A) ≠ 0 and P(B | A) = 1, thenA. A ⊂ B B.…
- If P (A|B) P (A), then which of the following is correct:A. P (B|A) P (B) B. P…
- If A and B are any two events such that P(A) + P(B) - P(A and B) = P(A),…
- Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) =…
- Compute P(A|B), if P(B) = 0.5 and P (A ∩ B) = 0.32
- If P (A) = 0.8, P (B) = 0.5 and P (B|A) = 0.4, find (i) P(A ∩ B) (ii) P(A|B)…
- Evaluate P(A ∪ B), if 2P(A) = P(B) = 5/13 and P(A|B) = 2/5.
- If P(A) = 6/11 , P(B) = 5/11 and P(A ∪ B) = 7/11, find (i) P(A∩B) (ii) P(A|B)…
- A coin is tossed three times, where (i) E : head on third toss, F : heads on…
- Two coins are tossed once, where (i) E : tail appears on one coin, F : one coin…
- A die is thrown three times, E : 4 appears on the third toss, F : 6 and 5…
- Mother, father and son line up at random for a family picture E : son on one…
- A black and a red dice are rolled. (a) Find the conditional probability of…
- A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5}…
- Assume that each born child is equally likely to be a boy or a girl. If a…
- An instructor has a question bank consisting of 300 easy True / False…
- Given that the two numbers appearing on throwing two dice are different. Find…
- Consider the experiment of throwing a die, if a multiple of 3 comes up, throw…
- If P (A) = 1/2, P(B) = 0, then P (A|B) isA. 0 B. 1/2 C. not defined D. 1…
- If A and B are events such that P(A|B) = P(B|A), thenA. A ⊂ B but A ≠ B B. A =…
- If P(A) = 3/5 and P (B) = 1/5, find P (A ∩ B) if A and B are independent…
- Two cards are drawn at random and without replacement from a pack of 52 playing…
- A box of oranges is inspected by examining three randomly selected oranges…
- A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on…
- A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event,…
- Let E and F be events with P (E)= 3/5, P (F) = 3/10 and P (E ∩ F) = 1/5. Are E…
- Given that the events A and B are such that P(A) = 1/2, P (A ∪ B) = 3/5 and…
- Let A and B be independent events with P (A) = 0.3 and P(B) = 0.4. Find (i) P(A…
- If A and B are two events such that P (A) = 1/4 , P (B) = 1/2 and P (A ∩ B) =…
- Events A and B are such that P (A) = 1/2, P(B) = 7/12 and P(not A or not B) =…
- Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6. Find…
- A die is tossed thrice. Find the probability of getting an odd number at least…
- Two balls are drawn at random with replacement from a box containing 10 black…
- Probability of solving specific problem independently by A and B are 1/2 and…
- One card is drawn at random from a well shuffled deck of 52 cards. In which of…
- In a hostel, 60% of the students read Hindi newspaper, 40% read English…
- The probability of obtaining an even prime number on each die, when a pair of…
- Two events A and B will be independent, if(A) A and B are mutually exclusive…
- An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour…
- A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black…
- Of the students in a college, it is known that 60% reside in hostel and 40% are…
- In answering a question on a multiple choice test, a student either knows the…
- A laboratory blood test is 99% effective in detecting a certain disease when it…
- There are three coins. One is a two headed coin (having head on both faces),…
- An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000…
- A factory has two machines A and B. Past record shows that machine A produced…
- Two groups are competing for the position on the Board of directors of a…
- Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three…
- A manufacturer has three machine operators A, B and C. The first operator A…
- A card from a pack of 52 cards is lost. From the remaining cards of the pack,…
- Probability that A speaks truth is 4/5. A coin is tossed. A reports that a…
- If A and B are two events such that A ⊂ B and P(B) ≠ 0, then which of the…
- X 0 1 2 P(X) 0.4 0.4 0.2 State which of the following are not the probability…
- X 0 1 2 3 4 P(X) 0.4 0.5 0.2 -0.1 0.3 State which of the following are not the…
- Y -1 0 1 P(Y) 0.6 0.1 0.2 State which of the following are not the probability…
- Z 3 2 1 0 -1 P(Z) 0.3 0.2 0.4 0.1 0.05 State which of the following are not…
- An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X…
- Let X represent the difference between the number of heads and the number of…
- number of heads in two tosses of a coin. Find the probability distribution of…
- number of tails in the simultaneous tosses of three coins. Find the…
- number of heads in four tosses of a coin. Find the probability distribution of…
- Find the probability distribution of the number of successes in two tosses of a…
- From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn…
- A coin is biased so that the head is 3 times as likely to occur as tail. If the…
- A random variable X has the following probability distribution: X 0 1 2 3 4 5 6…
- The random variable X has a probability distribution P(X) of the following…
- Find the mean number of heads in three tosses of a fair coin.
- Two dice are thrown simultaneously. If X denotes the number of sixes, find the…
- Two numbers are selected at random (without replacement) from the first six…
- Let X denote the sum of the numbers obtained when two fair dice are rolled.…
- A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18,…
- In a meeting, 70% of the members favour and 30% oppose a certain proposal. A…
- The mean of the numbers obtained on throwing a die having written 1 on three…
- Suppose that two cards are drawn at random from a deck of cards. Let X be the…
- A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the…
- A pair of dice is thrown 4 times. If getting a doublet is considered a success,…
- There are 5% defective items in a large bulk of items. What is the probability…
- Five cards are drawn successively with replacement from a well-shuffled deck of…
- The probability that a bulb produced by a factory will fuse after 150 days of…
- A bag consists of 10 balls each marked with one of the digits 0 to 9. If four…
- In an examination, 20 questions of true-false type are asked. Suppose a student…
- Suppose X has a binomial distribution b (6 , 1/2) . Show that X = 3 is the most…
- On a multiple choice examination with three possible answers for each of the…
- A person buys a lottery ticket in 50 lotteries, in each of which his chance of…
- Find the probability of getting 5 exactly twice in 7 throws of a die.…
- Find the probability of throwing at most 2 sixes in 6 throws of a single die.…
- It is known that 10% of certain articles manufactured are defective. What is…
- In a box containing 100 bulbs, 10 are defective. The probability that out of a…
- The probability that a student is not a swimmer is 1/5. Then the probability…
- A and B are two events such that P (A) ≠ 0. Find P(B|A), if: (i) A is a subset…
- A couple has two children, (i) Find the probability that both children are…
- Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person…
- Suppose that 90% of people are right-handed. What is the probability that at…
- An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15…
- In a hurdle race, a player has to cross 10 hurdles. The probability that he…
- A die is thrown again and again until three sixes are obtained. Find the…
- If a leap year is selected at random, what is the chance that it will contain…
- An experiment succeeds twice as often as it fails. Find the probability that in…
- How many times must a man toss a fair coin so that the probability of having…
- In a game, a man wins a rupee for a six and loses a rupee for any other number…
- Suppose we have four boxes A, B, C and D containing coloured marbles as given…
- Assume that the chances of a patient having a heart attack are 40%. It is also…
- If each element of a second order determinant is either zero or one, what is…
- An electronic assembly consists of two subsystems, say, A and B. From previous…
- Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black…
- If A and B are two events such that P(A) ≠ 0 and P(B | A) = 1, thenA. A ⊂ B B.…
- If P (A|B) P (A), then which of the following is correct:A. P (B|A) P (B) B. P…
- If A and B are any two events such that P(A) + P(B) - P(A and B) = P(A),…
Exercise 13.1
Question 1.Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E)
Answer:Given: P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2
We know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
Question 2.Compute P(A|B), if P(B) = 0.5 and P (A ∩ B) = 0.32
Answer:Given: P(B) = 0.5 and P(A ∩ B) = 0.32
We know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 3.If P (A) = 0.8, P (B) = 0.5 and P (B|A) = 0.4, find
(i) P(A ∩ B) (ii) P(A|B) (iii) P(A ∪ B)
Answer:Given: P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4
(i) We know that
By definition of conditional probability,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJQAAAAvCAMAAAAy7xNQAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjo6ZpCQZpDbZrbbZrb/kDoAkGYAkNv/tmYAtpA6tpBmttv/tv/btv//25A627aQ2////7Zm/9uQ/9u2/9vb//+2///bn2n6HwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC8UlEQVRYR+2Y7XbTMAyGnQJbgMEosLFAYCNdCsn9XyCRbFnyx4jdxM0OB/9Ys8y1X7+29FhT6n97Hg6MP65+pivp3t2nd87oOTYVtLd69LG5fjQPF/ohbOYbu/dTh+Obu4y50rsO+0ulDvULMGhsPpgv9tXumx3j1+eq2t3aX/Eb4/dq+qmOr7lb+pyzPccGBu8qWHKP0qC1NzXpU3092XeooZtxER6HPXrZPeno7Mx/6aBF9dWkQT9CGz49NDSbeXusaaf0C/SrlFXCKZ63u1QdudbrjWTF+PT76yt0lXd8iTP+d3GKQw2+mOlhpjtlFZK6lq2D0LgwgdfabV1RFQYThhKLOl49sjMkylqHyxhbEwlW64qaxLbwjnWYJsz+xUXRQVelRdH2DR/hFPUYkFN8eeLMMsZGvz+XqB5Pjwmvp0SZlKCKnCmaG8JbR72Js2GvrQic0smzhSQC0vFj3Qbn3M1Iwx7fwAeq8kUZzLz8gkIiKd3np581MonJGT155WFGt/ykMYJUlkfM/EyojZJU12NIatOoo4VEHjGDVc44RmsWVDduC2qTKAH5PGKeeJ9iVhGNCJDT6SRRAvJ5ViWfJbcjU53oJKhtRLmQLxCzvnZ2ihKwoLYRJSFfJrt5qpjqLMpSm46+gDxzoEWiQRO3yhP3KxBFVKesJqitRTmQn4GTFbrgQV62nO3T1NaiHMiXIWZs+/CdKwqpjaJcyPOZKrp9dNXzo4+cciFfhJj+SWSqU54S1AanPMifI09JquuMLqk9iQogv6QG8hJ8Ct9DfoZvco1yKuagYE7h+9q3BDgqgq2RgjlljSvfp0CUqANjBXMe39dJzSQKKuZowZxi1VpSeBx2Klow598D15AYstUtmItUL3PCuWK2VbzD0iJ13rwoIoat0x2WbiwqXjA/E1EuSzc5U8zWaMF8Fr6Ht21bMdPd3ymYt8lTQmWsYN4mowtRkUS5uVHi3+qkNOWWMJdzFv/9hPvU4jn/5QH+AHZhWrLwRWj/AAAAAElFTkSuQmCC)
⇒ P(A ∩ B) = P(B|A) P(A)
⇒ P(A ∩ B) = 0.4 × 0.8
⇒ P(A ∩ B) = 0.32
(ii) We know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ P(A|B) = 0.64
(iii) Now, ∵ P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
⇒ P(A ∪ B) = 0.8 + 0.5 – 0.32 = 1.3 – 0.32
⇒ P(A ∪ B) = 0.98
Question 4.Evaluate P(A ∪ B), if 2P(A) = P(B) = 5/13 and P(A|B) = 2/5.
Answer:Given:
and ![](data:image/png;base64,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)
……….(i)
We know that
By definition of conditional probability,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJQAAAAvCAMAAAAy7xNQAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjo6ZpCQZpDbZrbbZrb/kDoAkGYAkNv/tmYAtpA6tpBmttv/tv/btv//25A627aQ2////7Zm/9uQ/9u2/9vb//+2///bn2n6HwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC9UlEQVRYR+2Z7XbUIBCGyapt1KpdPxuNVrPNanL/F2hmmGGGhBXYDdHjKT+6KSXw8gLzMFtjHsu/4cD47eZHupLu1X1644yWY1NBeWl7H5s3D/RwZR+Whd7YvZ4aHF/cZYyV3nTYXxtzqJ+AQWNzSy/21e6L6+Pn+6rafXS/4hvj12r6aY7PpVn6mNGWYwOddxVMuUdpUNoPNeszfT3Zd6ihGbkIj8MevexOOhod+Q8NrKi+mjTYRyjDu+8Nj0a1x5pXylagX6WsUk7JuN216di13i6kKManX5+foauy4pc4M38XhzjU4AsNDyPdGaeQ1bViHRyNKzp4rVvWFVXhYcKjJKKONw/iDIty1uE0xpZOgtO6oia1LLJiHYYJWr+wKN7oprQoXr7hLeyiHg/kdL5m4mgaY2PrtxLV4+6h43VKFIUEU2RP8dhwvO2pp3M27K0VC6ds8GwhiIB0/Fi3wD73I9Kwxxr4QFVzUYSZp59QSCCkz/k5jxqZxJSInjzzZUR3/OQ+FqEsj5j5kdAapakufYwEBazRGM8k5mKWEcd4zorq4jZD3crUGM8k5pn3KWGV0MhB3YpiYuIGLnS58C0Uqgs/HdR9UUil/H2SvMmloTjl+Gkc1JUoxniZ6DYTLlQPQJ1FeRjnMNQi0aCoW+UZtgReEao7bAvUlVOJGHdCL3jQly3nlEA9sNHLEDO0fFgXgLoW5TDOt7Ciy8eD8OlTUA84VYSY820lVKc4paGugydhfIs4pamOEd2DusYMYfyiHGgW4FP4voyLy5pco7yMGVGYmxGvfUuAraLYSlPMzYhXvk9ZLLmMmYDvofSi3XB2pBa2MvA3yIhjasUpDjkbZMRJojBj5uBsd3rRjDguijNmEVU8I46LYmIw8L2NvglKFxrlPus5NZ3KghlxulO+qKIZcUyUsJVPX/mMOKZJsdUBH+9/pzPiWJfr/j2UMP+diK7mFUiEcvm+rkvY2wL4ed8CFFCEqvx/QKTcpwop+T+6/Q1SgFulZ/P05gAAAABJRU5ErkJggg==)
⇒ P(A ∩ B) = P(A|B) P(B)
……….(ii)
Now, ∵ P(A * B) = P(A) + P(B) – P(A ∩ B)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYsAAAArCAMAAAC+VVTjAAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZmZmZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLaQkLbbkNv/tmYAtmY6tpA6tpBmttv/tv/btv//25A625Bm27Zm27aQ29v/2////7Zm/9uQ/9u2/9vb//+2///bk2KQBwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFfUlEQVR4Xu1ba3fbNgy13Ee0tUkWd+26zlvaan2pdSN55v//ayPAp14ASMknPY34IUlt8ArEFQEQRDebdawWWC3wQ1ng63uhOmLBe8KTPHbpNUieKZJpy0KPLc+FWFAdXhbFExaQxzu8MiDqtiguP5GrqZkFWCj+mRsOSWTUTKG2/Es2UyxYFW82x92jLwzsEE/tY1WOLwuDofZP7457Eq8tSS48FL8GBklmqlwpXj2LLBasLvSMuuAoZrhorz7Vxv4N2Lmh8E67PyguAhS7BgYp18jCeax6DkcsiBNI26EEty80n4aLCn6ddsDwxKhukC9iWCh2DTyS0KxZYqx6eVygAckh5QJcFHABP8dHc3m3EBcCpCwjCye15fNSEGnhPRYKjr/0A3WGeN144faF4cL8HB2nF++NH5PsC3INEiShWbPE1N860u4FeZRYUKuh9jesMl28CrI5HEEV41h6XAwE8Vmai5oKUNZH0c8UIbHrmivQlrzt8BkyQbUnnHtH1w4etS+m40XjSBRwQa9BhDTX1Ox8MjLGs2WC1cWkb++p0sEb58LFbvJtEfoofDq9Bg6JteUMgQrWKHndxYKbGlx7w+20Id4EF5geM3kZZ0GXkvGL5ZBmmJqdWmn/zJykDIZYsP1Fx1GFpqbGEG/AhQkdkEIRoRufwZ2W7feCNXBIrEFnCBxfaX97RVcYEF4saKMrx8UQr8OFuoXizOM/4cm6pnJFej3t6YnsI0Dxa6CRYjurj5dc2h6J1xITz6DxQU9V+9+kQRHt1D7jChEP2pypi1d7TNmegxMJWTvhPu2E7TWwhq57HUtZADOwQwnHnsbXFZrgHm2G4Io3NmVTHwpM8TGlyRn9OJqD8dPNMecmyOuiE1T12h/OBlwYMVu/yd4YKxfmTWrRI9lhTNsUN1Fh8/T7Z1+emeQC94WkGDH6+q5cWLPE542wL8Kn9UVwSeNc/PfPE5Nw4WVCxnjgXJx2vj4WZcvIxaHUft9zAXbyFecRLgDmqd1alQsYY8U3gqOFuUh8esbL46cs/qh667JRzIswK3JW37SXdyF4jO8LBSdM3BfTwTswH/4yc3o10/mCI7YdA5352QSDGagdpJgL52X8vqgR3SZVjgtv9E7sprggX76F98WcF/0+5hI+CtVxXMCNRyiW2U9DkmX+Uvay3seLxK27MBeJT59j/oUfNYzdoJwLEQ26HVf5tSSEG8vOvjjtuLLQxLIX5mKOce91bienDdV2a3X76+S6XppSF0YOpc+XcIKqdBIM/OUevFcuhq8AhG4XfvHcDZ5MfwC/bMSA4uMW6pk4bA3k8Rv813ruPtOuSj+40dui0+Cn/oW8eXLYxjv17VemJ1AsyBgpRbk0ey+DvHCdNm7wa8rr79NL8o131fYd2akgFmTNJ1eOheoJLIS87P1F1ODXlManjY/QeMf0BIoFefuJleOh+lyEvkZ62cnIcyfgRTLfUuAPmvp5ZB+aWFCiuFA5CVRP5nzIGcq4KWhYvuMymPh4S14XigUlOguVk0D1N4ds2RnI+VPMuaR69FbfKeP108RwJob8jbx7FgsKdJYqJ4DqiZwPOV0XN8OkZjoPxjMKcS6MXI+52OJI09/TgrzSYuV4qJ7E+ZCTVfET/PER01mijhgKk1oOLlMkXNCCrNZy5VioARX2codddiryDHnb4Gevz+Ow2weNvyM728SCrNpy5VionsD5kFM1CfK+wc9kRvy+MHubjPSuIZkV5NROUI6D6n1/PuRERSLx0ODXlroD9hv1X39MO52ChvTjdPu9FhELMnqnKJdmgvMhp+nRkY4a/Fr4747vprCixjvo5pvOt8SCvNZi5Xiovocyl0vYWksuOxl5nbBa4Ke2wP/UMa60tPO5zwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Question 5.If P(A) = 6/11 , P(B) = 5/11 and P(A ∪ B) = 7/11, find
(i) P(A∩B) (ii) P(A|B) (iii) P(B|A)
Answer:Given: ![](data:image/png;base64,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)
(i) We know that P(A * B) = P(A) + P(B) – P(A ∩ B)
⇒ P(A ∩ B) = P(A) + P(B) – P(A ∪ B)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIQAAAAqCAMAAABFjAMLAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZmZmZpCQZpDbZrbbZrb/kDoAkGYAkGY6kNv/tmYAtpA6tpBmttv/tv/btv//25A627aQ2////7Zm/9uQ/9u2/9vb//+2///bgxhTzgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACEElEQVRYR+1Xf3PCIAylzinbdGu3uR92s86N2jr4/l9vEAigtbb1eod3K39ojgZ4eS9JKSHDCM4AG62CYyhpeBA8eQkPIouL4CCK2TY4CP64IqFBiDQmCgRbBqyPIjIjJAiIP7QclwPiEjqmzIsujUJ8zX7aZzGbb9o7t/YU6cO2tbN0LG/7yXmRQgndqZigpmGIdFKHxiwY3SuH8qaf1yNPpoTk9ErKUKgfnc6+iLuF1PTNMgQLxDqSv4SwWrBdGJVBq91YtDQWLM5eKZJCCio1yimcqVlSJk/g+POpKIH8vT2LKCYlRYX507fVQ5/pPUQQMO0U7BQ5sO1VjGPCzbIpYSiNmTVYkInfj2stXWYZaoeCJ9jH/bqF3XMqybUgRLp0kSOaDNXXmTkxZNrpzG7euiewEVIPe0Ku2+DL2dYlCM7ap4BaZOYoC6JKhAvZWXtePggk1DLBYJHR4zgITExyAsRRfU7IAf4IQl1DVJVqrmpAiFSDtDnRWY5qYvoFUID60A7qQZgS5UncLiErXnsliodhA8Bs4IkOtcKEblaZrGk5zu8TPiqVl5j30DGVaHJC/QGKQxCmbY/fNU/9dEwPUffO00hEvsCXi7Ma9Ov7Lbp7xhpzVnMK9XufKOcbI6azmjH072FbnGuF/R/StOMAAhkamBiYOKwW9+0V6itMfFJ5LxnLi7qzmmr63zz/A8fINX1R1WiOAAAAAElFTkSuQmCC)
(ii) Now, ∵ By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(iii) Again, ∵ By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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Question 6.A coin is tossed three times, where
(i) E : head on third toss, F : heads on first two tosses
(ii) E : at least two heads, F : at most two heads
(iii) E : at most two tails, F : at least one tail
Determine P(E|F)
Answer:The sample space of the given experiment will be:
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
(i) Here, E: head on third toss
And F: heads on first two tosses
⇒ E = {HHH, HTH, THH, TTH} and F = {HHH, HHT}
⇒ E ∩ F = {HHH}
So, ![](data:image/png;base64,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)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(ii) Here, E: at least two heads
And F: at most two heads
⇒ E = {HHH, HHT, HTH, THH} and F = {HHT, HTH, THH, HTT, THT, TTH, TTT}
⇒ E ∩ F = {HHT, HTH, THH}
So, ![](data:image/png;base64,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)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(iii) Here, E: at most two tails
And F: at least one tail
⇒ E = {HHH, HHT, HTH, THH, HTT, THT, TTH}
And F = {HHT, HTH, THH, HTT, THT, TTH, TTT}
So, ![](data:image/png;base64,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)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 7.Two coins are tossed once, where
(i) E : tail appears on one coin, F : one coin shows head
(ii) E : no tail appears, F : no head appears
Determine P(E|F)
Answer:The sample space of the given experiment will be:
S = {HH, HT, TH, TT}
(i) Here, E: tail appears on one coin
And F: one coin shows head
⇒ E = {HT, TH} and F = {HT, TH}
⇒ E ∩ F = {HT, TH}
So, ![](data:image/png;base64,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)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ P(E|F) = 1
(ii) Here, E: no tail appears
And F: no head appears
⇒ E = {HH} and F = {TT}
⇒ E ∩ F = ϕ
So, ![](data:image/png;base64,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)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ P(E|F) = 0
Question 8.A die is thrown three times,
E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses
Determine P(E|F)
Answer:The sample space has 216 outcomes, where each element of the sample space has 3 entries and is of the form (x, y, z) where 1 ≤ x, y, z ≤ 6.
Here, E: 4 appears on the third toss
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAa4AAACGCAMAAACyo55HAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtmY6tpBmttvbttv/tv//25A625Bm27Zm27aQ29v/2////7Zm/7aQ/9uQ/9u2//+2///brXlP4AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAP60lEQVR4Xu2da5vbthGFJaeJ1bSOnWydS7dpLDvJ2o632tX//3ElSFzmDAFwDhcK5SfQJ2uNwwFncBP4crDb9U/3QPdA90D3QDMPnN9987vpYncvf1PlNpGaje5mFd5GanKutdD59ttPxrKnf/4bSm4iJYzuVIXL0vdvtAvM0rnvtJuM3jUVO99+58t9/F7WGb/dPRv/7/R3WSQrPX98vd9/KYoR0g//WJaW6rvzdvzN5KxmpafDfvhMtzd9zNKh7Pnt4SvR2LP3agrDcqHz26+HQfD+i2kkfHi99/+af9udDv5+7mTlstLj/qfdw026FCN99svu4TZ5LivN13doSaGK493YpacDjhiEdHDe4dUfwtEzq8cXvy7HwVbidHBT0fn2+XR/L3+7Ez6Gb7vHmx+8E2X3ykuP7np3++CD1tK80Z2oorubrNXCrapwEdIhWj9JX8+l5/eHV7ZoLJU6HcY4icYlwuU8Lvra8bt7H640nlSkQ6uL4aKlx2g3Ky3VNxV2d0VIde8ipI83U1MPn5z0dAhzzVJAqv9/vp3cEuKgAgThuv/mUyw2dp7pU5QO7go+Z6UPP8fVZ15aMCoLDzXLVjgvVeFipKlRTv7IWsUpdW3M7vdT1A3hevzXm1TsmCavcriiC0jp481+/9LP3AVp3igU3lHS0+HFIS2NKOnxi/8Oy6pX9Qr7UWxtnLzuGNYOacwrDIbj+Dd46W6cjkS4RHmUxhGTl+4+HqYKlaRZo1iYku7O/xmWRmF9Q0nPt/vhJ9BHP9oV7zWN7utDdr71vWS5d927ha776HCVpGFKd1MYKXXdfer2JWnWKBampJMLwwxDSb0PfQMu3mtadq0PV+yiy+HCMdMwdx2fw89uctqTc3NGWh6Bxf/IQT5VuDLXwqLBatWHS44s2Xv10876YMU2bFwZpinu8SYtdAqLtPGn2X0qFm5hUTqNoXL+zkiLK1k5C4twCat56dEZhfWb2eo0zInpIdYB7xXXj2vCFntoGrn8r/nwh2lFg9/8tob/Y146/jI7D06gpWc3gzy4UboiLdZ3qn7877Agm6oz/rjMS90k/uCWybx0GKI+7T4Mk0RNOvwGlFsLa4Ilf8f6XYLzz24z5m8/Tqbx2ziTTEuTseuE2mWlx2m6iuEipA+uDm6h5Q1kpVmj02Q3VDFWLVfhrPTh+6G2csOAsHpy+22/CIfkrD6KLZ51wRp6cNzqkT983dUeb+SmDH7ze4bhj5tIKaNY4a2kYsNhZbxSuIYGCTvyb2FfBb/5reb4x02khNGwIx8qvIk0bEisjNQoE+HamR8C9eddJpcrN41T8hM/MlxPvFSX1z3QIFwNLtGjZPRAA183uISxsr1YA183uESPg9EDDXzd4BLGyvZiDXwNl+grw0ybagdRNQ4XQRZ1Eso0WKCb2obL/9SfAUx2EgqlFAmlpAQJNTNjJ6FQSpFQM6sGEqptuPxGmgKYkIuqk1AopUgoJSVIKG2GIKFQSpFQ2uoSCeU6Y9NwhT1RBJiQi1ogoVBKkVC8NF9fioRSt8qQUOrmFkmo5uGCxiWfNVEkFD6mokgoLbWTUKDkSCghZUkoIbWQUI17FzyDlViBCNcyziTYp2lL0k5CoZQhoaSSJKGElCWh5M1FPG+86YKbmg6GMlxQ8xQuA84kccWh4gwJJaUcCSWULAklpCwJJaQmEqrt3CU6ET4Oiv9hwZlQSpFQ6iEUQUIlJU1CCaMsCSWtmkioC/Uu8Wzcde0YLgPOhFKKhFJWCRJKKFkSambUTkIJqZGEulC4FMCE6GAdZ0IpRUKpwki5VJkkrSRIqJlROwklpEYSqmm44jSjAaZ8uLI4E0opEkoW5kgobYYgoUDKkVAoNZFQTcMVX8vIA0wWnAnZJ4qEgsIUCTUzYyehUEqRUOpWTSRU03CF97sQYKJIKJRSJBQWZkgobYYgoZRRhoRSVk0kVNtwUXjQ9HpXJ6HqvJjmr57MasiIE3hQJ6HCjnyNF1P8VdveNeyS9Df/589FrvZ5l+kRTi+02gOte9fqinShxQM9XBYvXU2ZHq6rCYWlIj1cFi9dTZnW4eorw05CaQ+0g6gIcusqckK17V2BhNIUEkFCgZQjoVDKkFC6LEFCgZQjobTVzUgopJAoEgqkHAmFUoaEUmUZEgqkHAmlrHYSaiGdVCeh0jRTQGvyKZaQz8BkH0ROqCKVs5jYCZQsCRXry5NQUbotCSUoJPnwv4j4iMe3KKVIKJQyJJRUkiSUkLIklKzhhiSUpJCA1SilWEqPb7WUIKFQypBQoORIKJByJBRItyWhYj6mcXSkSCiUUiQUSpmcUElJk1DCKEtCSatbklAObBT5cBgSCqUUCaWsEiSUULIk1MyonYQSUiMJBa/tr9trST/dgB+SuXYYEmqohJBSJBRK1bd6dqZCbqDJI5R0eDgsEgFZpUYSqkm4AhXtZ9o5hWQmobSUIKFQypBQ87JmEkpJGRJKS00kVNNwxfxO+XxMBhJKAUwMCYVShoSalzWTUErKkFD6Vk0kVJNwxbRS/v0upJAoEgqlFAmFUoaE0mUJEkoZZUgoZdVEQok80pmZa8hhKbJFFqY2MVZ3Esr7SKW/apZOSr3uoSMyheJ/Oj06FJO/DTsJNbkG4aaG6aT05gcGzE83R5XNHgpB9v7+vCszBrUjoTDHuzN1eiHOkpm9XjGvjO73634NdJXRA3q+gUXsGK4PtYSiD6+bpKI3VrYXG2Yvl3XQbWCFTwJ7p7VGDJdfqYljWe4a5PLtMaA8cIKDbQbp3bMwV4296773Lsqfly089S75UeGqm589Qb1sbf/qV/dzV2kwhJ41HwwH78lk2n91Z17+/nE/b7AnN2otC3nY2O0L+csu5OdvQsuF/PSrrD7eyXdWCRCsHX2mEzbXmni7YwW3OcyweiZU2ISqvgMG76/HVT2cXWQH15yrk5QC15SUANdQWfwGpy+K3z+pvhS4NrNjANfg3Il1I288sWYYR2OWc3kcIAOujX05tg4KXFNSAlxDZfmb3L/J3ioFrmk7FnBNAZBrApZym6d+lo7i00cbLqTwwvxZVAovXpqvL5XCCwvrcFUPM1R2TCm8mjxAUY8n3Woxnlg4NgAmhZeSMim8ZlJzCq8SquYqv3AioZSS4JqU2sC1JuEKg1dcIyL6RR1mqKUEuKaldnCtjKotHmYIUg5cA6npMEM8YmHNUCiTtsXTj+SJhdi7llJ4ITXmtOYUXkpKgGsVVG3pMEOUUuAaSrcD1xD9gnAtpfCaS83g2lxqBddqqFqpwn54V0YZcE1b3QxcQ/QLwrWUwmsmtYNrc6tWcK2Gqi2cSJgzagTXUGoE15q+MCT3N2CvgwPXpJQE1zD1FhxUt4CQFZJ2jfMDJaXAtXhlI7jWNFxypjWEq3AiIZznyx1mmKQMuIYhqXzLH2Yo6suAa8qO7TDDpuGSG1pwHCB1mKGgxhhwbfrFEA4hZMA1VJa+uVVP9jBDUV8GXFN2bIcZNg2X2NUQxwFS4JobetJJghS4hlIGXENl4ZtrDOL0xbSrIepLHWao7JjAtbbh6uCa/y10KXCtbbj6YYY+XJcC1xqHq6fwym01tAPXWodr3cZIVxk90MNldNR1FOvhuo44GGvRw2V01HUU6+G6jjgYa9E6XJ2Eyjj+aleGEg96/6baZGok1J8mJepbS+GF9Z3XvsJfLdyqstq2d/ldjRkQZCehUEqRUEpKkFBFgMm3N4865UgolFIk1KywgYRqGy6/kaaegTMkFEopEkpJCRKqDDCN8YpoVoaEQilFQunCFhKqabjioRrwtiV1mKHyuYNSF/JwDRtfExjOS/NKioSqh6tKQmmwA19VyErbnt8VKjBrZHYSKgMMW1N4ZaRGEqoMMLlmUCWh6uEipDYSqmnvCo8ka+FaOMyw5nNWaiahKgDTEglVDVe+wnkv2UioC4XrxWH/pVgY2g8zRJYIxrgliEpJCRKqAjAtkVAopUgoLLwdCbUDIMi53J7CayY1k1Bzq1YSCpVcCi+sL0NCaaubkVDT8jebE2qJhJpJ7STU3KqVhELlihRe+FqIkYRCq1uTUAAEcSSUlJIkFGJIDAlVAJhGpy7l4cJlApXCKxTekoSaAUHmFF7DMgw6JpHCC6UUCVUGmKYuEOnkm/g+VFhioJQioZTUlMKr6VIjpiIeU3gN9jFplyGF14jsJylFQoGUIqEKAFMciMMLXBkSCqUUCaVu1ZTCq2m4AgmFQBBFQqGUIqFQypBQBYAphGtCndyKaU5CKaNUCi8s3EmocSRTUBIcoUiRW83ycD3Jaj/McJqg3Ic4VpCBvtql8Go7GPbDDFPoxb+u9nlXtrb9j8080Lp3NatYv1DOAz1cn1W7aBCuBq83f1Yu27KyPVxbep+23cNFu2xLQftwrUfXzMpdu5Ux8dOjqdWVMU8pZ1ZeYJAd5VlG65N4EUr7OZDzu7qKJF72N3WwkbQIV2JfXJKNsGmNGNYyupZVzsi1kBSpkE5LGD2XDpRclo77g7DbU0HXhrLCqg1dS1si6KUsuibb1+NNPGJhdfeSZwnFd0ABw7Kga1mlJteqENnwpOPw6o9wG8figZLZTFwlbGy8XA1dQ6s2dC02TbRaqINoX43DFZ85QEIqC7qWV6ocXvn8X82lFLo2tBGRHNeUxCu+w41pu4pJvFL7Gl44X92rglA8Pg21nR0msIiuFZUyh1cdIsucYCBOsaxCZGVszN0kITUl8Qrtq/IUGqym7iUHsrVxE/NfeO6qMCxDEq+iUh7iWSfBZkZLB0qOfXb8FKwyp0+iVVMSr1AIpWV0LU3rOKOuDFi6SLh9xLDczB0TUxb4s6JS0LmctHygZEr2nLdKnT6Jt0qhayitoWu+fbVYGEroKWa2khjWuNAK4VrMiaWVYsXoVp2Dg+/GLPfJ50WjpQMlF6QUujbkv/3202DIL4hN6JpvJCjV33L3Ws3Ea+9rcYqIFRnXmyJl+eLpkyWlyISzkE4L2S8/2PlDJCgpha7lrC6ha7F9SS9Z0DWHdzT4xDkTnS7WF4voWkmpyLUKRJZhv4Bc46SFlF6zaS9n1YiuobSKrk2/tlosNPwPk6/Gc4kiz6UxrEV0raDU5FoVIpNGKwdKZjJxlRNo4ZJkUWpC19KM+bsYgyp1GBvJ+V0YcZ/ewc7vvnam0ys4zz/tPgybU2Z0La+ckWtxtyEDkUEWrNKBkq5NzTNxFRJoWdA1lNbRNX+99KaO9FI5idfUSI4ywf/TIzatjP2SAjAsE7qWVWpyzef3cmuXTDotMFo4UNIiHccdB6tZ0LUdWq2ia/56sRUgrFZE12DbrFGgpstQUNbnDpHhyWVIyGWTeIU/pnRtk9fq0qlpXuaz/vRJQtmOBPtz+bMAxulGXQXmPN94mXDZHw18/g+t1j+ge8LDoguFrV+25oH1ke5+/fw88H9wSOycuc0TIgAAAABJRU5ErkJggg==)
Now, F: 6 and 5 appears respectively on first two tosses
⇒ F = {(6, 5, 1), (6, 5, 2), (6, 5, 3), (6, 5, 4), (6, 5, 5), (6, 5, 6)}
⇒ E ∩ F = {(6, 5, 4)}
So, ![](data:image/png;base64,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)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 9.Mother, father and son line up at random for a family picture
E : son on one end, F : father in middle
Determine P(E|F)
Answer:Let M denote mother, F denote father and S denote son.
Then, the sample space for the given experiment will be:
S = {MFS, SFM, FSM, MSF, SMF, FMS}
Here, E: Son on one end
And F: Father in middle
⇒ E = {MFS, SFM, SMF, FMS} and F = {MFS, SFM}
⇒ E ∩ F = {MFS, SFM}
So, ![](data:image/png;base64,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)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ P(E|F) = 1
Question 10.A black and a red dice are rolled.
(a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.
(b) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.
Answer:Let B denote black colored die and R denote red colored die.
Then, the sample space for the given experiment will be:
![](data:image/png;base64,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)
(a) Let A be the event of ‘obtaining a sum greater than 9’ and B be the event of ‘getting a 5 on black die’.
Then, A = {(B4, R6), (B5, R5), (B5, R6), (B6, R4), (B6, R5), (B6, R6)}
And B = {(B5, R1), (B5, R2), (B5, R3), (B5, R4), (B5, R5), (B5, R6)}
⇒ A ∩ B = {(B5, R5), (B5, R6)}
So, ![](data:image/png;base64,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)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(b) Let A be the event of ‘obtaining a sum 8’ and B be the event of ‘getting a number less than 4 on red die’.
Then, A = {(B2, R6), (B3, R5), (B4, R4), (B5, R3), (B6, R2)}
And ![](data:image/png;base64,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)
⇒ A ∩ B = {(B5, R3), (B6, R2)}
So, ![](data:image/png;base64,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)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 11.A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5}
Find
(i) P(E|F) and P(F|E)
(ii) P(E|G) and P(G|E)
(iii) P((E ∪ F)|G) and P ((E ∩ F)|G)
Answer:The sample space for the given experiment will be:
S = {1, 2, 3, 4, 5, 6}
Here, E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5} ……….(I)
……….(II)
Now, E ∩ F = {3}, F ∩ G = {2, 3}, E ∩ G = {3, 5} ……….(III)
……….(IV)
(i) We know that
By definition of conditional probability,
![](data:image/png;base64,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)
[Using (II) and (IV)]
![](data:image/png;base64,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)
Similarly, we have
[Using (II) and (IV)]
![](data:image/png;base64,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)
(ii) We know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGoAAAAqCAMAAABspz7JAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkGY6kNv/tmYAtmY6tpA6tpBmttv/tv//25A627Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///bNgNxZgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB40lEQVRYR+1WbXODIAwm7qXupVvtts6Nbp11jsr//4EjARFrW0HuvOtd+aJCksc8CQ8wdhkjGShfPkd6BrrtlnD1E+gzzlzMN8VEUOoHL1DjqmS9LgSeEYHJNGohP1IAuH7zYkZ+P7rCIvOF61bMN15RfIxk/vzr2u1BMXH/7hPmiI3MVc4AD/i/FNlMAKioBqpcAiSUkbiLKUWdzRgrU5TJirSSJthfaqEKULluEVpp3G0n7bAcZY6RCxVJvzUP3kCJlAqG3+FpCaLLDA1QwYIJTMRC6VeFUrj9u1+9wbQqx7vNyszSxNYkuFBwnSOI08LQqDPqABotFkUuU1UDC4UGFqrOOuXh9osfCHboF4qkaVvquORJVduoMmFXe1BCbU7yaKF6YdtEOksuVEOJS6A2xsoYAk1znIJyAE4QSFaHoRgnsk3PsLZWfgT224IaudOBttmxOe1ineGH/+g0u96xmi560xOEa+QjWTH5pUsVIxfYFU1TkVo0woS06V0k1ykkr+t4tXDI6O3P3sTppCSK5Y2fSkYqO4cV22Wel86484q6E9V0olFNB8U9CYzPvNnq8ZGGIgQfMkMBj64bFRjtH+DIZxH3gQAcc/OowlQyCMAak5ZIPgWUOVamgBrHRZzXP1tOKndhpL/IAAAAAElFTkSuQmCC)
Similarly, we have
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(iii) Clearly, from (I), we have
E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5}
⇒ E ∪ F = {1, 2, 3, 5}
⇒ (E ∪ F) ∩ G = {2, 3, 5}
![](data:image/png;base64,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)
……….(V)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
[Using (II) and (V)]
![](data:image/png;base64,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)
Similarly, we have E ∩ F = {3} [Using (III)]
And G = {2, 3, 4, 5} [Using (I)]
⇒ (E ∩ F) ∩ G = {3}
……….(VI)
So,
[Using (II) and (VI)]
![](data:image/png;base64,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)
Question 12.Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?
Answer:Let B denote boy and G denote girl.
Then, the sample space of the given experiment will be:
S = {GG, GB, BG, BB}
Let E be the event that ‘both are girls’.
⇒ E = {GG}
![](data:image/png;base64,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)
(i) Let F be the event that ‘the youngest is a girl’.
⇒ F = {GG, BG}
……….(I)
Now, E ∩ F = {GG}
……….(II)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
[Using (I) and (II)]
![](data:image/png;base64,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)
(ii) Let H be the event that ‘at least one is a girl’.
⇒ H = {GG, GB, BG}
……….(III)
Now, E ∩ H = {GG}
……….(IV)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
[Using (III) and (IV)]
![](data:image/png;base64,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)
Question 13.An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?
Answer:Here, there are two types of questions, True/False or Multiple Choice Questions (T/F or MCQ), and each of them are divided into Easy and Difficult type, as shown below in the tree diagram.
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SIYAHSrB1aNcFn85ajTfNGcLDR3DbN1YwMJUsk2hKupCw8yc0Rd8+gMZNeGrWFHEmuIoOOUXPeGVi4RWndRVKmw2jvQnyyQHbK2rg4O5PJreIVHAOztIxFSIAfQlhuHJyQg7I3u/f+Dit7N7JWdffSAu6lS2H93vPwJEs2fy5HBAeEwPOpJwrkoMUVYwMn13zSpzGWxk1iGkWm+6MHT2jM1uQOkExHh0oAPjKa4h7M3Wh2RZRM8XV8pwiyzoy21uK6H/3DW/7G8lSwcGFcuniB3D83iVVwTMfCaEduWXkcXXsAgeFhyExRkpDgYNy7fx+VmjaU74aFRUhCwYX6fBROhKBmSYDsT641NztbZ+TIlQsH/jxGr+1RhEBt49qVmDR1CoFBfUz+ci6mjXlPskXsgprid+Eq4MtJBi+vx/LayoqjNPSbXHqWXV7FUWRNcnNydSNAfwZn+hNb6qqxUkluAJEgC/fNZqOEH+VWBmsbCSrZa4Pkz3SQUCk21aI14bHlCHpMDG499ER5Mtm4uTiSQDzzw3sj34cTxU6eUEbHkSL33FaSm1GtVhPkdnOVbAnHhmM5k2JtRP3WrdC8YXWEk/Xz7OFdlC9XEfWaNsWyBbOxZtVG2BC4cTOSq2UkQQoIi0azjj3wV+3GGDV6JPr2eAdlihdDrtwuiDgXhl4UJc/rlgmevr40JnuaF6ePVRZJsehNcl10UdEtKuatBbz9gsk3tkbz9l1w9mYAmpDreePEb3IxhbOk2VLsKjYmGFtWrYfntdMagxUq+FD8KypMvbayZyfcH+tWLkM5cmOtyRpRLUL+jSWBDvTzRSTx5cKlq/AN8KfFUY36Doc/AUewH6Uj8xZB0TzZcfToH+j3blv8de06DLRwSxQuqFmHsWjRsQsmzPwB69avQrXpE7Bi4Q84evoMxREWUywkGEEELNxCQqIRSu8j6b+adarjmyUrcOXCBVjntoOnTyA+bNpErgsMCiIFokZau15dzFn1E76aPhXNGtbEjZs3Ua9KeTjR1I4cOoj6NcrSIj+KYiR7BfLmkmwRR1hbtG2Fecs2YsfP2/Bev17Yt+8YnvqFY9igXtKvgRZ4dATRhtzEO+cPymdRZJUXKF0YeVyccOzofnTt0ghnzp1H3iJv4dKpo2TMOeHixcuoWKoUNqxbJwBitHHAs7sUb1qyClXLvUX9KhkND41E4SIl4EzYcYT66tSiBs5euoI69WqICgwOVQHqaALGUFrTfsQSS4VjytJOThcmzmhn8OCUEqeFAH//5zhFDIjBc5w5cwIFc9WT2586d1kGcuUK+Y91y8nrLNnyoUC2TJgxdQw8H16FhY0lbhMznlJdwIiPPkbbLl1QoWwp8ZWLlimPr76YgiDfe/A4chjzl2+SPvieLBBXb9xHFAVKt+/cgSGd28jfvJ9coyUeBY99Hhg5ZiKeePvj1rVruHv3Fi5f+Qt+fuQLn/gT7w3qj2+/X4CGbzfCldtPkS1nTgwd9h5+2rEfNShgVbxoMRQuXhZffDYZ9s42JA/Kz9ZDdolXmCjCp8emam+U/jp+4gSt+kgcOnYSvTq3x666TWXmfxw6QXGssrh5/RKBS2d0690PP+87hBU//oCe73RG9coEMpcP4cJFF9hQADqvsy0u/vUXevR/Dzt/PYVtmzagds3KOEHWpHtZd9y6dBZo15yC6uRshAfii4nT4PX4Jj759FN0adsYJ48dQt58BWCkQOVTqgfZvH07Bg0bgT69e6NK9RoYO26cgnSpjYhCmWoNKE62ETNmL8SUKREUZziBESNH4fTZsyhUphgM0c/JvT0PK4cs5HKVxLEjB9C2x7uYfPk6ls35Bs5O9pj2zSz069AOt25eRQwptEIFjDh59gLa9hqEKbcfYe2GdTh88Hd8MHYMSlWoieWrV2Eijbc3jalVh7Zo1rKVLDzO6nFKuHGrbthCcdelFIfbt+Mn3Lh1F78eOIxGBFzcuvfqjz37j2ENBTPf6UoZzUqVcezgPrRpWh3r6IsjRo+VvmtTAHVi3To4dPYYlm5YiquXLqM80XsQrRm+3zBShGdu3sZuyoA2qDUdf92+hzLEq2MH96MSgdVqAtVZ3/+IceODUb5OHYweMx5sGV6jAC/z9Nyp43CxiZTXPo/vwS84FC6UxGDdn4xBVN1wZ3RT0V1OfYYRivXoNxQ9BhqRyUmlX9lCKV2pBpYvLysRdqmxIBMlS+7CtEh34CQhas269dG6aQNa3I8I7UPQtF1HHDrggWOnzsGOTNoeFA22tbTEwrWbkS3XW6hU+i0TKkaSSdi191AK4JH5R9H+KOqbBSlzjkL4ee8e/HnqAqpWq4p2bTri8ZMn8PX1wfjJn5J5CuQgq2b69OkUab9DgJYdB/b/hny5csrPwYMHcfjP42Ru28n9HR1sxX5S7sub47TEBzmufWDwYIbGUOyqDvYfaIwQ8jXYMWEXNZY+Hzp6DEaNm4wQ8jk4vpC/WBkcPq5ZHvTtAQMGkylN7gH9d+DgYRG+MLqW3cjDZILrrWKpMhg5diwiItgCIfckKIASca74cOyHyEKgY2PDpnYsylWshmMnTpKlGk38BfITyB/5k8DNrCljSGVh2BRp2qoT/XRAQEAQ7B0cKcBvQCRlc44RmLC7Ex0VgUUr1tJ8KfBP9+c6jWkzZlJGMIwgiCxoO+WC5c9fEPsOsEVA8k3+AVsKUz+dIT/mrf07PcA/5k3sbXHz1Tpp16kH2pIl16FtC1y5fIHcEy/87nEQb5V4C8XcK+EEBfz11r/fYISH07hoXdVq0BRnzzJ4x7WCQ4ahBwX3g0iBOpOrx40rQyrVbIDzF1QKl9snX87BDCtbip0QjWnerdr1oOxONwSQSeVC4M6N61q+/m4hLIk/YRw/ouG2p+siKV5koaXpeY0nG4BoLqsijO4O0+/sOfNTUKlfvImyk9mKgqDxmlbxV6FqTfCPtOKUCze1WFSuXkd+9BYc5IUFi1fhvdFTyYPkQhvlRrhkyoJuPfrE+64alCXcK1aXH2kq9CKtfPnKpteMzok194qV6btx1/E1cfM2/0binybaabr4kEGEJ2sgi0C5GPbWcZkZBhhrchflcwoC/lOztrKJ+xP1x+ARr8VGSpqfl5c1/e3ZwzuUuiRX0dUeZymG0LJBLc3htaBUv+qLYykcJ0isCazTfTg3GNcMpsXFn1lZK1Aw0AQNZuOzMxubNQGH+Ugtza6zNaNDUlipV4hy/EYVJ5LdSu4zp2/DevTG4AH9yVrpgunTJiVajygAmvh0eRYSiNXBQ+aViEttReDBzVbAWDXmoQ4e8p7oaqnx1JZKK/RmRTEY85ZsAJIU4iX9Gp1CyjEwb6zt2FVgkYgiLdi/f190aN9USypx0ZfKAqmlnZDSb9rCTjpFX+VKycL8raT77z0m9brExsKcUkFJVcUZQya+a6Zs2LRjFwUmbRFO0UcOQPIwdBfRfB/oi3D2Vcb5KnSML8eqEFLGHROJTNlyUxp5D9UnhQgAiPQywmgVaXqIQIfCtDAHHnoaBRBzcUhMNLg60gAXt+wYMfw94Qsb1IK2ShI1Xr2IWL2YaKQVBr7YqNPu1XGgrxYNLy1r2v5gba9KAbhxbU/CvThJAba0NWueqap0ZmWotkHQMqRya7ZEOMvEuMFJBKMux6q8WYBVb2ll3mkUQP6Z5VIKzMSXWAoTlPfJqB0A3BLfMpX8IpRWGJj8M/t7j0mda1Kv++cxswKIWyRcji7LTVxTDohyDC1uIb3s/RJ+77UrA7G0OA2qbRTUYmhq/xXNld6rMnitrkhgM+WU4avIULoDEAblGMlFqz03KvehCZVEbl+FHBnfTTUKEEioqg21VGSTojgr5u7svwWq1fUv28xBJIa0vyijuA0nL9ttot8T14vBURuuSWwFMHWo0H4n5jq+/DSTdR7cWboDEJ3EFqY91ArNlbCpKpCMlt4ooIOEigsozND3ZstSNnNLE5vby2mNeDEUuu/du3dln4yooyTEfF6aypriU/ayjiJyV4nuyWeJgEQawg3T1NMhgLCFp0DixKljqEx7KDitpQj+coL00oKQ8cVkogAvGE7+8jELKi6gFhfHutTWev5EzmlRq1t+vZrNER8k1lAp+HvvvYdly5ahTZs2Ur6fsq6NDh36nBVQxn0an7Rx4KGPK20oynQBIOaM1DepjRkzBl9//TVu3rpBhWVFRJrUATIZLX1TQLMiJRYSb9nE421yqopjx46hF+0a5tRxhw4dqODxjBRNpdxpn2ZumZZZShIY6helISFPNwCiLwpm6rRp0wQ82Eft0L4jdlDxWb58tJeBtwGnIeKm74WcGqPnehNNJIWPCkRMutaMt/HZnHSm6/ENdlH4JzAwEOPHj6e4Wgx2794tQPLRRx/Ja6N+rkkykyJeQFSsKX1rX9yNEp2RRpO0FFBN8wCiawH+zYDB2oErRd999135qUOltxMnTsTKlStTLOiVzPKT0V0qUSBhzIOHMX/+fHh4eFBV9HLUr1+fytynUNXsAPTs2ZOOclibSiNNP7dNFwCia4ubtC+mR48eyJQpk2iJEiVKiMm5ahVtkKpWDUOGDEnZ4Ff64WvGSP+FAnqAlK2MCRMmiEzxnhJu/fv3p1PzdmIdbURjIGFQyWj/TIE0DyDmqbTBdNbm1atXadflUQEPbhvo7IOmtLt22LBhaNiwIYoVUwfGZLQMCiSkgA4c/DskJARt27ZFwYIF8eWXX8a7dC6donfjxg20b98e3nTsQ0qlc98EDqV5ANGJPHv2bNruvA996XCX6tWrmwJc7Kd+Sjse//zzT9EYu3btgpOTk+nvKZqOexMk4D80Bz0Yz7/Z/Y2kzXyffPIJctGZGubuDady169fD3d3d7FIFi5cSOdlqD0+GfIUX2DSBYDwLtgPPvgAnTt3xuLFi+PNgBlftWpViYuMHj0anI5jS4UZnfKpuP/Q6nsDpqovflZEGzduFBDpTkcucgBVlxd9mmXKlBFwmURHKZaiszVYtjLk6e9CkOYB5Pnz5+CUraOjo4AEm5PmkXSeEjOWrY9ff/1VXBnejPTOO+/IbFM2l/8GrKr/2BR86SAodl2KFy8uAVNzq4JlS7dEGFBY7lhhcdavAR33V758+f8Ytf7/dNM8gHz77bc4fvw4HXS7WuIb5maoOTiw27JlyxY6qyE/3qdHObCQ2Nvb/38KZFzxn6FANG1YY0simI4O5AAqy0pi1oeueLguhK0Vdpk//vhj7N279z9Dq6RONM0BiLkG4NQao3+rVq3Qrl27eNoiMV+UAYNjJZzL79ixowRYM+IhSRWFN/c6XdGwxfHjjz+Ka8LBU27mAVJdpswVU5EiRfDVV19J7I3daL3+SHd5/usxkTQHILpJ6e/vLwGskiVLChDY0bNZktI4JcffHT58OJYsWSLWiHktiZ4STkpfGdekbwqYK6OTJ0+Crdm36Ml5XDeUsJkDgflr7qNPnz5iibBy6tSpk1gkuuWSvin06qNPcwCiM4/3I7B2+OKLLwQ8dIYlBfG5CIhdnnF0Lia7Pc2bN89g+KvLSrrrwVxWODYWEBAg7rDuopjHP/5tctzP5MmT8fvvv6Nbt244deoUPZQsy99icemOQMkw4DQFILrpyBHyAwcOSGFY69atXzgQykFULgaqWLGiaAwOnLE/mxFQTQaJSYdd8OJnC4TjGHocLSmyoGddeMpsueyhE8M4kMp1I+zKZFizaWA7f7yNcoT0zKQudPp67dq1Jf7xok03W1lDcNaGzU+OifAuS46RJEVwXvSeGdenLQqY8/ganbrPC75fv35izeotKYvfvPCMv1euXDlxjb+hZ81mzpxZAElv/1W5ShMWiE78cDr9mTc2cfuJnoOaNat6lm1S3BZzwdA1B5cns9nJ7gwznxn+In2lrWWRMZr/RwHzYjC+9sGDB2jWrJlsfdBTti8jT+b35b0zrOS4eLFr165yfojuXr+Im/3/5pJe/p4mAISJxeDB7sZZek4H59518HhVQnLEnTfgsRaqW7euBMAy2n+DAmwp3LlzR6qT8+bNm2yTXkqPt2xJT0bkeAhXQOv1I0mxapJtEGmko1QHEH2XLTOZ4xZcVcrZF24vaxaaV6GyhuDt/pyOYwuEYysZ7c2lgC5P27Ztw7x586TA8O23336hIPy/UYf75x3gnOFjV3vOnDkYNWqU6SsvK7PplSOpDiCM3vfpeaAcp+Cg53Z6wpgOHq9CVHMQKUzPJtXLklmgFixYQM9UTfWpv8r0Mr77DxRgeXr8+LEoC65e5jiaHkBPDqLpLjAH99mF4TIB3tjZpEmTFDyAKDlGnjJ9vNZVlFg9BlcHcrqVqwM///xzeoBydplpcsQqzPvg3D8DFbtHXObO27TNfebkuF/KsCij16RSQOchB8/ZdeGtDTnpiXXm9SBJ7eufrtP7sqVn1bBbzNXPDFacncmWLdurdp/uvv9aAcTcstCZzedQ8sEtM2fOROPGjZOdgObCwwHa3377TXxXdpcqV6780m5Ssg80o8NkoQC7LYsWLZJgPMe89IB6cm7J12WKT8FjV6lFixai/Nid+a+11wogCct/jxw5IhYBHwbEZcIp2ZjpHA9hrcQ5fS4sOkHPRM2wPFKS6q+3b866jBw5UgLlfFCQDh78mzMkyQ0iLDuc5eF7Mnjw9n/e4ftfaq8VQHTXRF+0nGblKlNOs6ZUM9/fwPcoWrSomJwzZsyQ3ZZcG5BSZ1+m1Jwy+v07Bfhh3Hw6Hdf6sDVgXvOTnEpCV4LmI5g6dao8eJ33y7A18l9yZV4rgOimH2sDRu1bt27hs88+Awc5k9NPTWyBmQsRgwaXNXM1IZu5zPSMeEj6gyVzmeFjLbnalKtO69Wrl2xZl8SoYm7J8BhcXFxEjtka4cAqW7kcpP8vZGReK4Doi5gtDt7YNHbsWPFVExYAvQ5R5ntzxoddJzY98+TJk7Ff5nUQPpnuYQ4et2/fltQ/Z0I4K5LQ0k2mW/5jNzwWPlaT5Zo3c/JpZvyb25sOIq8VQJig169fl1OeOADFByPrTc/fpzSz9f75/nxUHR8VwAVH7MMmp6n7uubxX72P7krwPicuU7exsZGMiO66vG66sPzy6WY//PCDnOjO+7E41fumt9cOIFyPce/ePXAAlfcTmJf/vm605qzPwIEDwYfost+aWtbQmy5kKTU/BhF2hffv3y+ZvNQCD/P5sVXLcs2lCXzIt4ODQ0pNP030m+IAYm5qcmSczTxO3XLFKTfdn0wNN4YLjDjlx+XzPDbWHFzynBEPSROy+bdBJOTL1q1bxV1o1KiRpOb19jotyYT3cnV1FZnigkV2p9giSSjbr3N8Kc3JFAUQ8xz8lStXJDrOzGaNn9A/TE2isgvDZ4awH8sbpRhE9HNXX7drldIMfxP6Z1nhvVO8QNnq4AOnUgM8/omWHI/hQjYO1vNB4FxKn9JJgtTia4oCiL65iOMevPmI0VnfUp2WTnTivQ28+5cBhHdZsgYxP2D3dbtWqSUMaf2+5nxgDc+uMPPKzc0tzQUrGdx++eUXOZv39OnTcg5JwsPA0zq9kzK+FAUQfQCc8eBIOT8Qive76IKQmlZHQuKwZcS7gdnkLF26NEaMGPE3KykpBM24JuUooLu7fLwgp205dcoxrLTWWL75PBq2tFk5cZ0In+/Lgd43rSU7gCQ01VhDcLkvF43pW+kTK8ZJbcKycPLjDJ8+fSoxGs7OcOVqRktbFLh48aJodT7fZejQoWkS5HXFyIdi8X4ZVqAcsH8Tq1RTBEB0AvKuSEZfNjH5NGzzlpasDx309KfccTyEN9yxW2O+GSttLaX/3mg47sEHAwUFBckJc+wSc0tLspSQK1ztzPuvWCnxAeF68uBN4V6yA4jOUM7PsxYPDAyUgq2E+2DSIgHZR2Wtwelc/uGS9xUrVqTFof4nx8RncHDmhV0XtkDSS+OiyZo1a4rl5OnpmV6GnaRxJjuA6NqAmc0ngbELw/nw9BSI5HMv+eAhdmk4sGqeIkwSVTMuSnYKnD9/XgoQCxUqJPGptGx1JJw8B1C5LuTDDz+UsfOxiG9KS1YA0UGCj85n040r8fSS3rTMcB6b+RPK+GFUbDXxsYrMeLak+HCajJZ6FGA+8P4SdgeYP+mt8RPxnjx5IlXPXDmrW1DpSbEmRvNXAhC9zkMHB/7t5+cnjw9kZnNKNL08XjLhVm+OmHOpOxeX8VPuNm3aFA9E0jIgprfF9W/jZRnjalNOifJJcmyBpLeaCh0kOB7IhZS86Y4P++aHdnNLzyDySgCiM9682IrPieQydQ5A8hPO03PjQCr7rGx6rlmzBoMHD07XzE5PvNAXFT/EiU1+LsjS6Z/e5qGDBJe1//zzz6hUqZI8HY/jOem9vRKAmFseTAjW0ryVmd2W9u3bp2va6ALMZzwwo/ng3Fy5cr3Ug67SNSFSafAsW/yIUpYj3imtPyMovWlr85IFHjvXQfFhVt9995388Ov03F4JQMwnfujQIdESfMAsHyv3JjRmOKeg+VR3Li7jPL6Xl1e8k63Sm0CnF75wTIofav3w4UM5ta548eIm6y89bi8wlxPOynAVLYMiB+n5PJz02l4aQMz9UE7V8kEu3Ph3WiwtflEGmcc4+MFEfFAvl0+zi8apXfOdnxkg8qLUTfx6czpyIJ73TnGxGD9dkFtCizd57pryvehbOszvxFs6uEqVa474aE0+lEh3ddJTfO2lAcTcNOPIMm+p5jJwPpEp4e7DlGdRyt+BN0gxUPIBRDVq1JCNXBnAkXx0N5eZmzdvymM+2HVhM/9NazxXtmj5WM1BgwZJjEc/wzU9gQfz5aUBRNcIHB1nTcG+nV6qmx5NzKQIKe+74D0NXMjEx/jz0XkqE8XaMSk9ZFyTFAqwpccgwm6xrpXNLZCk9JHWr2G5Ycvq8OHDUt/CO3b5cPH0tuHOBCAvo019fXykRoKf+sZAEncOpJ6aErandV4mYXx8qnesHA7DgWIuDGrTpo3k9flQaCBGCTrN9U2YbRII8gqXENomQiVd886ZO0esWQaRWrVqCVXTY/un9UTqxmS58nk0K1euhIeHh8x3795fKVCf8yWnmzhdX7KzJH+NTn7VrmXJp9e8FKItomEgJhujDYrX9CuGJ06LyBhLb4xARGQkRZCHgx8M9dlnn0jRleqKl5ECDlp29C/1GEtf4L9a8BX0/dRuNIwoGgvP0RBLM7OgccoYacw0eB45z5hpwUOW+VgYEENag091//TTzygNNwEjhw7HwiWL+U9MNfq7UWZnMkMT8FS95X+1H/rFo1DmC9OKKWagnizINORrooR2MXQD/huPlb9BXJCetP9Tm5rKAhMaRdNrNR/hOwVCYWEkmtJ7ttTo77pAxTChZSZqTrCwFECeOJGqTQsXkZoPkUbuXObL8ybKMJ+EkNy3kieRM2FcKsqWGc4p7ipOx8kP0UY+tVTbOvga4qcFjXnp0sV0nmsz9OjVG3/8/quatsbVGJqWvmpIAoXGas6Kpkwbll+5o8gwyzP3r9Yh04XfC4nlU/67QWQ/OZrJAlGD4ZvQouL/RCpoEdF9eHjakE0z++2P37Fu/ToMHzqEjtPvhGi63hDPjteXC/UojFa9p42mjceM6XEsY2Ir1LRgYGGh5IVOc2Cht6I5TpgwHteuXcHi5cvQizIzterU0mikpqgEWkePhEJtRgOmi774tAXFJFTfYE6QcOhWjYyV/6KsHQUeCRAqDRCXp6HEWi1oDR5k0bBkyexJ6VgwsDBI83zpNbdOVLAXRYrpO8pS8MZGhnABZBF8ba46+bRFqGSK75U2ZIvnJMNhIGU5EB7z6zg5YCBU41Ug2rhxU/Ts3QerVizHEso49SdrBLF0jaZnuMhR2bb8Q0uWZUahgwCs6kXrn2TWpGBYOcrf+Hpdrrkn5XaLGL0i3cxiIHFMEObLKAk1RVAZULQBEK89/tiHHt3eQckSpTGGtior+4S/w7pTZyi/ZiFgXRRNr5QopI2mlqUajYJGHrta/SzYQloBkhgWVGEUCTMjPGtVo0GyTYeOHEP7Du2xbcd2OarAxBgLshykKY3wd6uLrjStfQUiqqkFZtT+qAuFgLlOO7FGuGlAkiZoGofEPDYyVIVurFA0SVJT5IVE9FNQEiXXsOXBbcmSpThMBYirVq2k1GYTWRisKU3Ki9kh2lRHXP5AI5vpOp2Or/u3NiZhJQ9KWYwWYoHxD/NbLXA1fpY4HRoVDiz8/jt4PfXESKo36tylIx3K7EZ0jIFRFj7TIg5OpSeFCrJGWU6ZVrxEdWnmNwzSvPZUU+pGZJEtWVaOIp+v1kwAEs0Ml9sQGNBkRUjpM1n4PHkWCgNbGcBHVJnp5+uHixcuUaQ8Nw0xgq630rSu5vbELUPtVdrRlorQComF+DJVNgE1IguDFbkV+ZkOSmDFHIyOobhPUezeuwclixbBJKoq5ENudEYJk8Tq0gFTvf4bfMrHSuAUfDFIqfvpClf6FIbzolSuCwtJ2ozcqhmKXGvUY8qZxFTQUS2kaBZwTW9uXL+RYgD90bpNa9k2IFM26VSdE7wgmEL8H0MrSaZmmqurX5c9ZkItjd/6ewUMPC5NZEzgyO9j6G9Gdklllallxy6M8nGMUhbAAfp8+QrQ6X1tqPZoJx1X4KLdw4LkI1JoaupfZFELF2j0Vtaecvfi1KMaj3Jd2O3RQT0O9LWbvNQvcsgENtUCkcamu/rNP6xNWKh1A4QH8tlnn8M/MEjAI5YnxoAjqoInqut1+kx8dn2pKs2TNkxNDdXZ/KMxq7lbidRrS4CYy/62siEsyIRkJsfQG/ZkrTRzsUSRwtiwcQPy5s6tCY3ONn6okM4P3VKge+lKkzUK9ys/KqjGb9h11NnA9zEyaMgHanmIcCh2CYxo5uFLMT55v6TGqAOfABx/EhOBuTO/wrnLd5A9Z1442VviGmVXuvUZhKYNamnyEYuCBfPT5jJ3zJo1G7a2dnHmNT+SUmila0umA4GsyX1RS0NZY6+uTZODJl6eD+DrH8JCAx8/H1jTnip3OnpAyZgFgvy9MGzIaNRs0AwD+neXT88c/wOrN+yCnYMbuW/zMYCyfY0b00HRHdvgr6t3UaZKbQwf3FMUNK+pGMFgXl9sIStlYmDwEgVIUkSLMTwkCDNnzcLVW7fxwUcfo1ypYtr648sVFCVHU1BI/W1avQqHT5yEo6szMdoez548R5Va9dClY2uNReLdyesmtBmIW3RsuAyEUVFpcza1NFHiCzUAEa3Ni0BQJm0wWsl7LMKCg/Dk2RNY2djBx8eXFi09Q7dQUdjb2SrBpTnN/fIreIdEY8K0cbCmmfr5PcOSRavw7Lkn7dYdg2AC04mTJsDO0RmWUbEICovGpKkTYM3rnZnKjGe6EBorA0PpV/FWxYdV0Y4dm9bg98PHUKN5K3Rs0kgYw2yOEgqzBmKNpazBOOsmOcTg1fpQwKg5rzw3Al/GRIPRFg7WsVixcgW+X7oZnVrVwZARQ3Dt+lU0JwDRg56V6YT+s2fPkTypccQpMNFJCYSdZU3BvFJuImhK/l7Rn086FXRlqwxBJdv824hnnncoINoCRsc8+PCjEZRh2YPQkBhMnvoJqlcuj5jIKOyh/TCOOQvIPO5cOYnGTVrh81lLUCBfblw/fxiuTm4kV5NgawjF1zO/Q4R1JrnWggNGIjm6ptNiI6b1pUuFAdZ29nBysKa4ygp07t6fAKQ4ZnwxHWUr1UTzRg0lnpcc5FIRGRpSJnsrzKOCluEfjsOg/j0w84tv8Mcf+9GVAITHy2xTS98CkeTbWREYGC3MznhkdUn9xEVl6WotOCYGprhBJhWfdF6lwJWi8DUZCCIA6dejO46fv45RH41FsM99nDx9AaPGjEOH1i1k7j9v24rb3hGYQgASHRaAjo3eRt5S1Sno1wq/79qN98eMx1Q6us7V2RHz6Pf9x4EYP2WCthiU6a2RTjcwhOrRYq/pmsACebO5YP68b+Ft64quTRpjz9a1uOcfhQF9etF1VmKDiCWig7QYNyy8cQKdAuT6/13q5pQMR9lMEnymV5nd1Nb7IsWKIwtl6pYvX03AYkAI0f3IocM4ff4sAXIA3u03CMWKFKD4agiWL10L3wB/+PpSmUDrJrh47hzuPfBC9Tr1UDhPNqxesx6FS5ZF7+5d5B7aGla0TRVS6BYYUKpCaRTI4QAf66wYQVW0gwf1QcXSpdC6fRdcuHSGQMEK5/66DEsXV6HLrh174U0WSyZ6LlHdWpUw97MxcHHNghp168PF3oBTp2rA4OCMMJqlnawxC9y6cxV29i7I7OKG4GB/CiPZwMbBCTEkmxGkvKztHODoYIcC+dkytoINWXX3rl/AuPFT0HvgUJQoWhg5c+eBrSWv0VcjmLK1qY+s2Z1lQoWLlZOnxn0z52uEElpGRkfhxpUL+G3ffoSFRqJ+01aoWK4kXRmDvbt+xrVbdxEWFopKtaoj6NljMlHvowCl4apWKo3NG7bC0SUrevbrDWt2A1KHu39bACoppNK3WbLlQK2qZfDH0UvoO2g4CuVwxLA+3dCpTSscPHMeVUoWxQrKNsXYOAiAXrx0Ah6nLuKjpl3RmCyxBd98iUdPPJGvcAk0qlGR+qpIIRIbWJpZiD4+XhJ75SP4IiIp3Utv+LgAi9hI+IcEwsbOFbZWlsibPwfsbG3g5JwZwXRs3ycTJuGJITOakjmbLRNdY8MpQNZ6ytUR3r+iAPx/dEjqFQreVLSffX5xyIjKSu3s2LQO65c/Q8WqDTG0fxd88/WnOH72Eu13+RotG9aHB9H/kMcOTBo9lGojfsGhkycwiR5CdplO9LeOCSO3+VMsK18VdXPnJWt5LSbPqyHSJEAssbpXWwhJnWVCGVbgbfJXJcNkoAUdTUL2lAJb2Y32GNG3LQaOmYODJ88jK/zQq/cA9Bo4Cp9P+QhX/roit/7j0D5kdojEk0fP4BsSiZ27f0Yuwt4hQz9E624DMO+bKXj68A4mT/oCNpmd8edBD1QqXwlN6tfEgBGjMeGLWWhWuwLaNG+GwmXrYdfWlWSxcMyFcj4xkTh+9ITc5/7dB9i+cwd69+kLW6dXP+OGJFJJepTUfgBH9nvgyf3LJKy2mDphHLZtXIf5c2dj7oJFGPvBe5i9aA2uXz2FnavmYcCwKdi9zwM7t2/EDjLLWtWqgDH0uMr3p3yGlk0bYc3qdWjeritsOPhFQBUtlkwaaJqq4pw8NwMFRY2kGbzJhWEAGfxubyxYsR5bftqNPK4d0KNrN7jkKITd2zfg5Kmz4sodP30Ce3/7FRfOnqF3RuzatQvWsUGYNvZjWNpnw669O4h3UZj/3UIco70OFlFhuHf/Mb4gC2XYkKEoW6EW5s+ehu6dO+LGowAcOLIfNtZsZcSIprly6SLueHrBmMURPy5bjc6dWqFksbfEitPCv2nFoKM1pRcuKAuTXVZxL+StWtiFixZDjqzlqFCqsLwfNHQEmno+hcf+w8hF2vAyHWYdwfIRa4Unvs/ABWXDBg9FlYqlEep7B3ajxtJjUa/gXqkSaNy8Hbq2aUGhew5HqpyOGEGvC0N0ETbDDQERuT+NRXcx5X0szdtFvvGcYiKdOzbBs8eeePSQjza0wNv16+D7letoQfdB1RJvoShZYV43A8gq7oqY4CfwpWs9nz6Trr+cNA47957Hw8dXsH3rJjx/5k3Zv4p4/vw57tHrYsVKIaudAY8eecn9VD6RYkZWVmhHBZ+WGIXaTdti1Ii+ybYIaQUpqhspQsi3y5HZDeVLl0GkBFqAhlTgUoYCXJeu3EDWHJnx9PhVPA0IRVhQCEIjQmhj2Ur0e7cnypYpCWcHS7yVNweZWPcpiBOCku5lyC0YKYIUy+k7zaVJttG/ZEfiSXHgV/csaFwWBk7Pqg9cs2WGDQnCw3ueFOArApvoSJw/+5f8rfHbDeR3vRYt0aJpM3hdPo0lG7aj34C+KFswL+YRRQ+dvwQrQuPftm3CeyM+wBminUOMLxb8sBTl3UsiNjwSF6/dhatbFjJvCZh+34xwor+1QfMoI8JQqVp15M2VDRa5C2HqRPUM4RgCJBVvUu1lqodfkmT/+jWxPZiYEiSjdypwIU2vL8r/Vim0a1QBXp73CQiu4syxI/h89mJ8Mn0KMmXNDKPPY0QTXT6f8wOMDlnpkQiz6XGVG7F+23Y0ruWOTi0aYgdZgo+eBqBT+3ZarkFJL8dBVJwttZrZvbn2QlL+XKQoFEBAoMrl5ciRBZFhYXAwEK+tlfsfEUoBV+JqeDDDIcjijySFFolgem2MJGVCityaFAu3Pw4eI0u3rCj6du06yWd3rx2luBwBqSX/CziQ+xLBroncmRutaxpLON0nlqJpEaTIkrNpEku30upW8rxVDF1IK0aFBuP2jevweuaFkWPH0CnlXZDNLQesrCiSHBKGfqMm43FAFObMnotN69Zg0eoN6N62Mbp1aoFF63/Fp19nR6UaNZHZnsw5NmRTlcHxSaaq+PgzLuhhAnPOnt8rlRIaEYRwep05Tw55b29lS9kB9bfoyHBhTHiEqvWIjIpk9iEkXAmJFZUn29o7yjUHqdjO0sKKgDcnWTL5MXvOfOoghPxgCoQRXaRvCnZZ29qrMfB4KFAWE6sso0j6zIZVualxVaoKGqpawtRcNGbDEpFW1ofyK/TKUAtyg1UWKjTYV6qWR44agbIVG+CUxy+4evUOAW0Mbt+5gthQa7I0AvD5V19iPAWkm71dBzUbNsO5a3cIQCqRWzkYVdt0gkvuIqg/c6rEgyScKtZP3P3MR5Xir3XvRbNo5X6WtrAxWFMhXDTcNHN75fpfkD1/UdStWJ6Y6kvjpdiXBiA2HGmneVgRUHAz0El+UWQNMxwYaK1xoN3airdLAC6ZrHDr7mX4hUfBnr721PMxxZOsJDlsQ8AQHhGOa2Tl5i5ZTK43UmKAwcnG1oE8jChZhxaRgfDxD6IAv5G2Zti+cvzMBCCqtBi0EBgRgXkzvsIjn2eItbHC8SNHMWH8RJzevx+RBB5WNPhvvuLH9r1DAb4+yJm/MH7/zUMApEevbqRZlmL91h04dvQP6UsvrI1XGpHi3E38BvpyVFF7Jr0RljaUWyFXjrNP3Pb8uk8EtG1bKmgSFpA2Ia0hTNGCG3bk58p7AgyeobWmAXQrhv/maGuNqNgQ3L9/jwCkKB7QYy7ykDkbSwAVHck6Brh1+4GKiVhbEgxRXwS0llrf4aStbAmgIjizExRIW76d6U4qSGnudqcSKc1uqxXGCSWUK8HgFh0dhut3H8qelsP7duLqmf146h1MGy8ro27V0vAMmIhtW7ejWfP2FEy9gjOnj8Pbzxtjx31MD2bKhMnTZqB/15ZynyrNWqJY3kK0Aa0nHEn4xaLVQeu1Aql5xFhjhBZH5Hf3CfAuXL0HZArFbnLrjx85SBm8WKxfuwZZKLN3fP9J+FEX1ymuyLu7z1+9Qd+KxunjR1Eijyuu37qHR3e8cJvOQYnxfYDn4aG4duk8WTEBlAWcjrYd+qBciVLISxmbHt06oU+PTnAvVACzv5iKwOf3YO3oggf3r9PPI/x58hz1HU7W85+o1aMV3IsVxNzPP8Ht63cw8/NP6YhOBUyv0jQAicW5i3/JQa83LpzCtOmf4k+qsmxJeeg6davjsMchLKTsQIWK7qhW0QsXjh3GE/LNJk+dinLuZSlv3R+Tx78v4yji3hAVKlRAtcbtkIfcISmQYgInpPurjPoVv6sUBg/IgCBizOETZxER7U/z/ANnrEKweOUmfDlrLppULQ/f509w+fYtPAmxpLTtc9y6d1emcu7kcYqAd8bVm7fpXSiunD+HotkdyDW5Ac/HQZSJ8UTXvoOxZP12tKSUZYUK5VCoZCl8/xXtXK5aGYvWbUXf/j3h7f0cURF+uHPtGhD8AGGkRW6ScEVGRaFG9apYsnYrWrTpgInjx6BuzWpq5ho5ZRqvSIvk+LquiPXskKRWuTaBwHnaZ19hGu/soQUfSWXqXBehtyNHGsa7fRRZKM1btqB0LtU6UE7XikzxsJBHmDNzMeydc6Bi4wZo17KRJkqq8lKltVkZvE57zEyYTdYHzzmarAFLLFy9XuqLggICULJMOYyZMBUujqSc6O/2rlmxZetPxDdLschqNG6Jn2q+jcyZyb2JCsfUT2fQvCiyQ7Qy2rpQHO4nooVBXJvGzbvhz6MFKZtzA9nz5EXj+vWIADHYtH0nTl26JHLVq3sP3L1HIYSISNRt2BxbKlaDEx2KZbTNhK07f8aJ0xdRoWp1ikVlSQ7W61nXaHQfPATvUjQ3PDKCsi2hmETPRTFo2vbE6bOmm31Cr8LoAT/vdOxA10YilF67aieWr1ryLRwyZUOewoUweEAv+Y7K03OMIVnG+8qdqAXHG9/YZTAiIiIYXXv3o8U+FNFRIQgKD8PKtZtQvkxZYU5YcDAmz/iSixoQQK8dXHNiJe1ZsLC2JyaFoVbDJihdtS5yZnGmBz4H4+Mpn5IwW1KhXTDKFHen7NUv+PX3Q4ixskenrp1hIItl2sy5qPZ2CxQsUhDZsrii46lLFIOKgrVTdiynvo1kckbGROOb+d+ixtvNkSt/EdQkMOGFJYFJlYpJI/ARnyVqPSkqq1SuYjwX4zF4qLIwcmll+PEzc5ZkWUjQjzM4Wj0AbyDc+8su3H0WgR/Xr0FWe6rPYa9IwEMVKvL90kSFEY2pULGS8pOwMe84ylimXHX6iftro0ZN413armPneO+LFY3ri/fQuJevJj96YykuTGeL8I+0goXFwuNWhAKyemP6FKS++IdbXCnj34b6Qh+YgqhWJODcbKys6Ucl4IRLstuJ/8K1keyRWVGGRmkRG4ruWtGPajE4QOXcuw6ewJyFC1C8YD4tQ6yESd/Wk1y7AF9olgku5mo+2TtAKixT5pzo/W7/v3XHRU1G0prsnvV+V2UOpOXPT4fi1jW9bd3enOGx6EXpMb1F0H0KFHXHQPoxfUY6iqsy+/TtZ/qsRMkKptcVKlbR6EXxkUw56YwVNTbGC1XErQca0ob1YRq4uC2shQXh1A8FqnWc0wqe6c/63gxGCKW1Tbk5DmcQhsiuD20zmZV9Tuzc86fEqiwpyB1LwCr9Sj9qF7Uq6Y4Dq1eRjf//3YQ2n3kgRAGZUpn8ghSVXE62mAZ2PGr+u6VmpUXKRrlYeq+UgmyZYYNKigal7lau54CsZJuoH86XGsW2VwCsomLqR/bCULA9lhMDMgZNXngpy9VMZI1eyWC+EhfViEU0ZZB8Tw4y8nDYz1TbcxTW64PSAMGCA4js/yp18e2yFZgVZYAz5ZfVCVPUr+CPHrSUy1O9SSmvlg5Vz4NRgT7ZSCdMY23B8+YAq9oUqDA7iuiiNtrLfhplOYsAC1O4tJj+Klkn+RZ/gzInlGFhSyyGAoYc/GQolgg9f40+U4ym+4k1rvggwsPrkMxcXdOq2ILe0oTOVWTThyQv1AhFw+lri6miCCXyxCIvc2MqqXUj31AlFYr2qsROgQLFFVXjYhq5ha4/tc112r6a1JOtOK4oXrLeZQBQ9bn6lnpVUKkWudpsqEOAxlgV2Wfp0IKbei2NTF5WaVyxpl4QoY6d4M91e05twlQWqiZd2rYBdZ+EpSs6+17mt8YaEnbxI3WMInyTe3H2RGclLya1aISBGqLKNKR82QBbO2dwLFlhB/en9KUSprRTSMZjMXnMsippbCyc2gJWm9V4z4ECkbj9uXr5r7YytDiKMF3zxdU6USgvgiIqhQVdaRBO7ykgUFvC9JoOkSyp4lSfqYXFv/XMi6qr0MtQeWOjgqk0gMgJxqHVy5pGxmOUmclQNZdGE3oWfzUtnS7MBF6AKjynNqCpYw30c1FU3YmyYOQEFS4P0LAltcmhK2FlabBloc9HWfU8E91aUEqb6RJL7nAEeclsZfEmQUUnXypAjKKO+KwdFggFCRQPobDBMzrMK3dOPnyI+1V305MUbH3oO7iVBcTj0MBJ9ssomUkOyYnbzs9M01BMK4fQxII3dLDEa+d9aHflxUHDigMK+lyMKvLlxfc1mbGalkgzws5D4/EpvFULkFWBEmw1d2Vx8KYkQXPTvgHT1iH5nmC/KFptGev1D1KXoKw1dS6GuhOfmaKCfUo7c6BNH4Fis9aTPjbT77j9Hzrn9SI4NYvUbfHAWBPMuM80OrMFxvQUxRyDp88e4O7jEFSl0m/dsvLy86XtAE6U2qWMFMu7sMSSAt2BYFPfjU6EMznDOlk1QNKKc1OXEIrN0uLsAaU8lHQpZSJb3ogG0SSHlvS3Mx6b0W3gVMxbsQFNaqhYxubVi7Fx614qC7BECYqbjBkzkraPWOLW1fMYP246chXKQ08ICKRg/0yKv7kpwGALWO6l2YT02lQRotVgJQdomBPZBCAxJMxqkpTKDQ6FLeWI9S3svJ1LF9iw0HCyNKgMW8x7tSCYKKFhQVSGTaWxxHxlqplWqNZrcg89hWVFBDQO1UOD/bDjpy0IpuBIDkonNn+7vqkY7f6dv7DP4wjsqE6mU7tWnBgWOYqidO2GdesRFm2JVu3aUR2Ni1Ro/tPG80StCRmHGe0SkjWFyZBc3VtQynzGx0OwdLMH157DL9AX3QZNQnWqNI0mOk2f+Am8w4Lx+IEnxtDrahVVsG8nPZ943badFFSOoqxCG6pO7S6WmsmNMe2vUvo5rTSTjjcBnc44HqdyP3nxhQXcwyTad3X95gOEUOaN2+1zB9B70Gis3HMY1bIEoEDp+nirWn10rlkEfalGK2fd7pg9czLKFciFsZ8Xx+pZY2Tq+pKLs/ZSniJxe99o4J4P7mDR4oUICYvEjWu3UKd5Zwzr/w4VnVji4e0rlJenR1U6u8DBLT/GTxgNZ2syp6jCbdqUaQiKDkeglx8mfv4FCubJZcbHtGJmv6hoaYwm5Pa6dxltW7dF3XZ9kc3SG/369seXP2zCmAEdce+vo2jRZSSGvD8E2+d9Co9D57Bg1iTa2OSHnt0HwCZ3fhSwC6U9CiuxbvduFHAjJ0/LTL3oiNL39bG4euUqmrTvhS4tG4jBl79IcZnSj3M+xferd+DenXOY+fEgdKK9H1evHIHXlX3o3n8Q1uw9iHwWD1GlXg8Uq1ANjapQUNsML9IWdPw7l8QtZbtA8zdW0daQ3Nlo05vBj1xmBTKHf9lCGUEresZSceS29UQmsrw8DhxDk/zhVG5xEy0/rk1XGVCpdAns2OOB0K8/ku0PqQGf8QBk17pFVLewDXf+uoAvRg/HB4N7oFnbNiiWOYbOJmiPIvU747Mpw1CyQDHYZc6DqSO7Y9b0D7Fq1zlcPvkbhndpTSnR0Tjw2zqqtlQ+oDqrkuarWVWpMssXXXlsDbIhqJmDnIIrWbYiPhj7EdyibuH9STNx7CRtgqKT5xZ8OQWB0c4YRk+wK2Hpg4b9vsDIcaMQdmoLNm3ZhZP3n6O88yN8+U0ZrNr8OyYNaKniJVqA+kWHlp6v57hGqUrV6HkovAC0FhOMbWRlFC3XAPaU3atZpRwmz9mGy3ef4eZvOxFusEGxsu4oEm2kRRKOHbv3C4DoIqUqc7UzbVNjBb0gQ9QQlVI9/OtG3A2xQg/atbt4d2/Y26qdy56kiPkaa47L0byNZPUG+XvjGT13mss87ahSlZsFlVkE+z9HaBTvn5KzIxK1Vl9wiC90ebwYSJOOPVChRS+qNDXSDlNvVK3TDrld7XDr3C6cOHsVfT5pBRfnbChfLB89vf43TBjSEtsoN1+h0XtwpPLthnWqYMWEDbj2yA/ueV0lSpBGts+9EFHiQE6d2ZGjQBksXrVe+ti3dY/8btuOi6Bicf70VeTKV08+c3LJTH6LD+7dvo0n5y/THgY7ONsSs6m02YWOS7hM29IBrqxkYI2Lw7zY4NLr1VQ6TfXXv25ajjsn9uCJTwi+mvk1cjob8NTXFw75tVJuAdcABNOGuifPAym4T1vV2bGPsqG0eixteORNaLx6+H91zJ+UBqQnBUWg5+tzm44D/Qw9h3+I0+cuylwObFuJ+zdKwEC7s41kUXAQnwPGsVQ0Y0GhASvyBAQ4ed+WNJq0FkfjdzGS8FCl8a8LS+MAhG6bt1BpOHt7YuyooVi+ZQ/5YAfhZGPE2fu0U5L+bqvVfBipjt/Px1vKa328Q1DQWjHfio+ri/RCAKElCEAUlznQqEVXXtesXnmNqTJp8xOteSo+D2+Q+zIJvUdNQs9mNWheAVT6z6lsJb36SaiRdAxCMO2V4e0Bwmre4k0/UVS0JozW2PzKw0xXHRgxZtpM2DtloyyCN2pWrYVe4db4ZdUs0p50HKYpRctCQllA2U/CWSv+YVHiNDdvHtfqjiQoqx3DqYNxepAvGSNHwaww7AM6T5jWx4PHd5T8UBlBCToqMxy+VOAYjmA6+gG2VKxJVaXuFSohN1WfZqHdDs/onBRuXj5+KFO+ooQSlFzxeTz6EZCvRzjigqicZaGbO9NhJh9R6W0U7YkZ0LUNKlOtvgMdTiKhUj0VRBzliDqnnWI57aThHWceRLtq+Wz2AVS6Vw8cqldpv7FGU6BgcsGiAjC417uo2bEfls+erk3BAq4utnhGxwFwU66JLcWJ3JCFPufTtCXyztYlaRRnN7JQNGrEc+LTPkFecYREy+gI2lRYEE5OLtRXLtSrTOfFnDonxnxxeq7QMcrAcAui81GMtllQpFBehFHRXjht/gqVlHoIwqiIrEKlimosEkhVNSVKUaUXE4Qd+1hkypQP3Xu+I1NZ73ONPolEIfcqqFO5AvxzOaJE/ixYs2AmDjvFoHCpiujWohqsnG3oEO822LN2MWoWtMM12jLx5fR+ksmRzYWaoo6jySuyLQlfj2eBBAX70E5SFzpkJxfaNa2OWRTguXKXToKisz/56BF/ipJzC/H3Q9ESDcmdyYx8eTLDK0AxP5AOwbHLnB/5cusPx+GpUC5HMEM3u/QCmCSMLpUukTy5uUzSLOZ/MR6Pgg1Y+9lEejDyfRw6fRk92jdF7TrVMH2d2up/58oVZM9XFFVpi/4jnzJU5u+Ne3SWQzZHbwSGRqN1W1W2bNSKfFNpeqlwWyJmdDDepnL8pj2Hol3jOvCgYyH6DZ8oKdtm3bph/ftfYf/RI1j20z707NsDBch1tqvXArkyzcHGVQvhEvMcb5WugG6taijIYPBQmUtRSaqeJC3UOSeBvGyUy5ogSaPsksGpAD3d8TNYRwWSFUK7bnO/hd8OeOBHOjjJn1y3HT9vRS5nOiuW5vjtyg34ccUqXCC354dlq0j+qsrxAWrbgEaP17jNwQQgTPplM6Zg08GLWEvno27YcRBZ85dCucJZkck1O6rXcMeudUtRMWcsbj4Nw/xZXUjD2qBVp86Yung7RYd7Y/Oeoxg0oCfyuTqYguR60EiRNT1YH2qkJhCh116PbmPh4o3wjjSiLp1rGRoWAfeaLQhAmqH7iPH448p4fDBuAm6dPo2lP8yFE/Hyreot6ACmYZg7fTQyUfDvowmT0Zxy/Er41R3SEz2SsCz+/RIrFwlCX7zxCHv2/Iov5y9Bl05thA7t+owgxVOANjN6oG6zLvTAsoHSV/bC5bGfNjguX78JXrRTeeeObcjlYi+qSK+9ibM70lCh4r9Sggvm9HVACpbct87vxm1rUPOJpc1yBfHxx+pYTPlEFxzaqf1uv4RbL7RVpoeCXuPGwngWSPMOXXH+YRAdErQc9lkL4+CR7yiIyntkaMfftp/pzNSF2LptN+YtW4NW9crTrCIwfNwXcMpRDJs3bUDTLn0wkg7NNS0P0Q4ai7WKzPSAIeKu6bMgzjm5Zccvh44LE6PJXWHf2zFTFjlfIWfBstj18yZcvHwDzu+NRIGc2YTbBhs3zJg5D9euXKICMiuU1M5oYGqYtM8rr8p01AG5t116DgSfYvr3ZkTTVm3lJ2ErWKIspk0rG+9j0bXaGtRCCq8xbJgMNDfTowlVqnr/d0WrMCdxBawDUpzcvj5FbQYg9KyTcjWx9Mea8Sik9rSQ/541DybSNn/zxjEPI22+6ztQgYbeeKcrN1ONT3pAjX+SC+KFrb0D8uX/+/mR6nGfRB2jHcqWVUKutrvRlyhjYEGxomL08C296dpSicLrY3IyiHxGFxkUSJQCZgBiHr3lNCPvelTHEPJz5Vj4uURWnhVmeuYp7dOgP6iohlo6apGYO/npLQOTMCOo2SKyT0Wbn2w2VO9lUxhNWnZ7yHZ77dlglF7TC3X1577oGkT1qNf9ZkhmBgXSLwVMAKL0ptoqrPxLhgVtA508Vk+lNQVGZAcp/agn3PAGGDE3dLNfmVTxNWx6iZHHZ2XCOagNX6ppxqbpvZZxkSyUAlABVJm4DqiKxvLdOPMs/UpPxsj/8xSIV4nK9Rr6fjB5bqmuJ0nFKhuE07L8OX+NrRM9xsEqWIFJYqZ5+gSPBFAShxwafih7hH/03Y0KE5g+EumIo4Upxaj3+R8LoP7nl9mbS4B4QVSW/Tjx1xQl/zI9GFhPt/NVXM/PhDHz7OM7+SaqpTdvPynj1Z6CooBCwwN1XoOy0uLRJTF3JSk3eXPlLmNmbwgFzFwY3ZpQZrY6XE6TcpP2ZUuDzW/z2at9HZr58YaQJSnT0Mt14uyrv+2yjefu6G8ykCMp1M24Jn1QIL4FImPWzWtVJqwdM2Ayy2W56Hlm09rRHuwTH1k0bZw+CBF/lEl3ukzPRBGyqVqEOAcljpZx/ZvbeBlgkh6lI2PMcRSIHwNJEPjUsEK7OhENaib/WkjxDaHt/1/YcVdoBUzygTkV/q2P/9//G0LIjGm84RRIxAL5txlnCP7fqZNBkzd8jWRM718o8IIAkkHLDApkUCCDAom4MBlEyaBABgUyKPCiFPgfBR6EbLI3j10AAAAASUVORK5CYII=)
So, in all, there are, 500 T/F questions and 900 MCQs.
Also, there are 800 Easy questions and 600 Difficult questions.
⇒ The sample space of this experiment has 500 + 900 = 1400 outcomes.
Now,
Let E be the event of ‘getting an Easy question’ and F be the event of ‘getting an MCQ’.
and
……….(i)
Now, E ∩ F is the event of getting an MCQ which is Easy.
Clearly, from the diagram, we know that there are 500 MCQs that are easy.
So,
……….(ii)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
[Using (i) and (ii)]
![](data:image/png;base64,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)
Question 14.Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4’.
Answer:The sample space of the given experiment is:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUYAAACGCAMAAACWltPCAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZmZmZpC2ZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtmY6tpBmttvbttv/tv//25A625Bm27Zm27aQ29v/2////7Zm/9uQ/9u2//+2///b59nSwQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAANrElEQVR4Xu2d62LbxhGFQefGpvGlUeu0TdU0oRxXsWpR4vu/W7H3OYPdOQuIcBQT/GVah8PBwWIB7McZDMP22hzYHLhwB07vXv23ZcHt6/f4p2ciHtbLY+FoOF3/5WP7o8fv/iX/+EzEQzuP334ZhqckvdjF7+Mn734YM8iv9O74J/G/p+uK+HT3drf7yqs6xB/+3C92IW9fTCOPLlbyOO5348vLeR6j6HSz/9oNIBDPd/F08+14MN9/EY7oh7e7+C/17tZ/V3hVxYfdj8PDVfgwF7/4eXi4DtZQsdvGfdCCuJ7HcV8OnI7Q9/s3/wtblcSHl7/Od/G4d/Pe6fob/9Hj6/e3xUb5TuysuvjgItzu/DacVzwMj1d/jzbKMdNIWthI8xju9z8mz5L49Nv+zVwfj3vvn9iFwkbnSzK1HEBt8ThOg4194kOI3SE+fH8fbRTiRh5yNNLQj1dh+GAex32aLDrtPF2HTUlZSuPCSM9j0482/2qKhyE6M3SIH35Kp38qvn/1MX9nETfykDbSPOJuD9tVQseZuNNEN3qC7z025smxbWPehAMTP17tdq/jbMvEj3/9pSRYxE0bX+7jqW50huRx+OI/44nxTUikiOMx2m3jIc3cZUJsHNTiS4QCxeUYKhk1xcPdPn4pEfuo46679fOFsLEe+vTv8VQXz14s6dP1brzMu4vHsAidjqo+I0/XcW+dZTSmKR821pgB0qFAhsy9u4BxL22jETrNbiR03P6oknsofFnnKw/eHhvp3Hj4Jl8T0enOJZi3lUYWsw6dG/2mp1MHySPaGIe1DB0nuz4f03joOFM/XuWzV+O07q+77r2KicPRH+d3JobJW4gbSR9c6LiHaOhw8IZhiKHLGZxbGS/zynXj+JFwlkoHaD5nuauq+J/i4BVif9l18hsRrhuDrCo+ubnrIc4oTOw3I+UhxPXQg5vuH+IFCE96Px5BH8TFbkz38UrcbVAfs43phuD0k7uV+vKfIcX8zm3HGDdZEu9iUHwIE5i30Q/MqK6KH9z3xBMkFfuBG27vIHI96YcfxizitRRNeji6O9ifw54SST/G2zHqoBekEzVcA/u/PF7hHOtHQfpPeQ3cFCf1emJ54X72pMUtMfWy2KgXS27yPZIPEhZL8n+qlZWqOKvXE6+XdLotoQ6q0TiOx229sZjmJ+/ulxiN3Z+5COEsG2eJL8K+tJGznJkl3mxsObDZeBZnNhvPb+N2phaezhpgUvxMYN+sNNYjg0ttjPcaAu+FfWORQRQTMqjENhmcpGGSQVQTMjgJXSWDS22Md74S740mCk5YgWwoJmRQiW0yqNIgZBDVhAzq0FMy6EbPQhvTooPEe8gJK5ANxYQMPkHMyKBK2iaDKK6RwSfYCBxIch5CBvOiYZyeTTKoxSYZBDElg0LNyaAQ18ng4tEoVr8L3nPWEDKIYkIGUUzIoBRzMijUnAwKcYMMLj2opY2Qh7CxBtnkCrRbCEoHVJWBSDElg0LcQQaF+rhnZFCIG2RwqY0CsuH6YM3GppiQQbXyaJPBIu4hgyI0J4MydIMMLjzFlNEoFv3VQV0ZjSgmZFBFLpC8NnSFuIMMTkJbZFCIm2TwyTYKvMfnRhQTMqgi22RQi+1fTUxCW2RQiJtkcKGNeVITeM+ffPMBXINsKCZkUIopGdRp5F9IVMkgqBkZRHGDDC60Mf84K+G9HjKYWaAXEzIIYkYGMbLfnfIXZYpQopqQQZV0gwwutDFBtoz3esggigkZRDEhgygeXTTJoAptk0EVukEGl9rYxe8uiQzOYDHGCs/Fk8GFNm5kUC7hLj2oL4q08I3dbOQedSg2GztM4pLNRu5Rh2KzscMkLllu40YGNzI4GV9PrBlcOhoTGSzELmRmkkEQMzKIYkIGdRo2GQQ1I4M69CpkUBC70URCBkHMyCCKCRnENBgZBDUjgyr0RgYbOPO5kEFZW0PJIBbiEDKIYkIGhbiDDGZ1DxnM4vXIYCF2fPV7QDEhgygmZFCKORkUak4GhXgtMiiJHWUxWmySQRQTMghiSgZBzcggiNcjg6WWr2WjLBTMeM+JCRnEyKRmsPy5hwyK0JwMytBrkUG/1FzKiRmnlmJWM4iRbTIo0ugggyppQcuqcD0n3SSDotRlzi0P/GpCVmXTX00IMasZLFsXUjNrBpWY1FPr0GbNYBE3yeBMG1MRTZyXgdj5TTXIoBabZBDFhAxO07DIoFLbZFCLG2RwoY2ZtaVavg4yWPBeBxlEMSGDKrLfnW0yqNQ2GdRJN8jgTBvL1BHGZSZ2PWQQxYQMopiQQRT7mc+oGVShbTKoQjfIYOo9Mn61u3n8EmvW1HQpLj43Mhh+wJngZ/n9w3jzmKv366cbvOCHrlAXTgZzXXuY3I6vjI5Z2BtoW2+EsRbPrWQguo/oYl9+hXRBijTL3aay79a2P7yd2bTngkz0V7XhvHKze/GP0kMjtQ+JZzzXB2tz0RwYx9ACbhjbBXzVbMno1mQ3Hw0f02gcJXfRUH6ivqwDtmNr4Qow38vFi+PU2TDEEf17OgJfliQ69/g3dzxj8yttBCxIbBc80p50MzzeoXwc21KaXXnkL4Bn1Tw+E/F6pZd59Wn48N14H1pO1bVDEn6Tn8830HWPNGXNSwbjmi1tyirFtCmrELvUSVNWoeZNWTF0vSkr0AUynYlL9NRnVbBMd64vLVpr/U1BzJqyopg0ZcU0WFNWULOmrCp0HbCqnWj7WLrNlHFZ+p/S0ktZEhmbSLabsj5BzEovMbS0sVIviuJW6eXMhTK1bOvO3rn/qd8FNmBVYrsp60RsAVYUM8Aq1RSwSnELsM60Mf3COZ+zJcsEG2v9TbXYBKxabAJWFDPACmoGWEHcAqxPtBFYprJxQocmYguwKrENWFHMACuqCWBF8TqAFVlmy8a8hiSbpTqxAVinkQ3AimIGWFVoG7Dq0KsAVmSZZDROxBZgnUY2ACuKGWCthW6WXqJ4vdJL6LrMy9LljRAFrHDXxAArihlgVaFtwJrFq5VejiMQMjJLL1FMm7KWyLT0Uqdhl16CmpVeonid0st8w9ABWMMFkTvXdwBWFLPSy0karCmryIM3ZZVJr1N66XaVZ5k9gBXFBLCimABWFOd3breJzqn50RKQNGnKqkJvpZfrdpJdWDPY0zr1kpqyLrRxK72U6w9LKxHIWtCl/Xmz8Sx7fLNxs/EsDpwlyDYaf2cbNzIodsDi0SivG13dIr6MJx/+jmIgg5gHKb1kSS+1MS4VCqwWfLTIIIoJGVRimwxO0jDJIKoJGZyEXqP0ElbgGRlEMSGDSmyTQZUGIYOoJmRQh16l9FJlJB7eWHvyoShypGRQ2Uie7QiRGRls20iTXqkp62QYWGRwIrbIYEXcJoNabJPBto3i90xJhOKVSi/bNlbIYNuZHrFFBlVkQgbbNtI8Viq9BKzmF0/LUy8nZHAitsigEttkEMWMDKKakEEUr0MGB8BqLRuTtROxVXo5EVullyBmZFAlTUovVehVyGC4yGnUDNarGIWYl17iM2LNxzWWNBgZnCZNSy9THmuSwVwTokZj7aGK5cGIo5iSQYjMyCCKGRlUoUnpZfrzWmRQYjW/ly0yqMQ2GQQxI4OTNEwyiGpCBpV4HTKYsVoPGUQxacoKYkYGJ2mYZBDVhAyqpNchgxmr9ZBBFLPSy1QR6SOz0ksQMzKIeRAyiOLW4xqfeE+dFyQu/nGNC5HWRgb9Yz4j/Fw6GrfHNb6XS4PLbTzLqvHnEmSz8Sx7crPx09s46xfOZ0nvjxJk1mjcbGzt1s3Gswz4p9j4+SPWbotnPuZ7Jx4t/0zqKWelMa/4sv9Xc/NsjLVpfieJJ0zdyGp2UnyZ+KSLQIsvpZgWXwqxX2sKt2eCUrWSbiHW4jmGju8ki3+8yqurHSNY9sbLxZcFOY4RWPGlFNPiSxCz4ktIgxZfSnULseaBgqHzO7GHFtuYl67vY8cPvxeONmKVfJIi1ieIGWLF0I3iy/xTcdhC+dFSpXvcm4XoaoQKvpi+WyJHXLit0MqJ2EKsFXEbsWqxjVhR3Si+FAX7wgb50TIc5WHKj2oxk6ZV+kk9ttXBsVK8nbrv+med+lczsoVYVWSCWFHdKL5M/41i+a4MFCjN5z4WedpYQI5+ajcQ60RsIVYlthErihliRTVBrChGxBr3/bwTtYSAuZ5SIseWjU2xWXypI5vFl1LMEOvpGkI3EGscKCjGd6kVDBJMPhqH3AE7f4k/04vWMsZoRD7prz/yoTx5lOVE7I732JmKiBlirYWeFl9i87b4lXXE6oDNrFeeYdFG0XjGmBuRTzr3x24r8TWZGydidyGwD41CesQGYq2FbiJWFCvEGq4W551gwiXN/mu/7psm4DA8a6Ox8txLJSaIVUamiFWnYSNWUDcQa5n95RbCJvjdeXoXd+6s8Xh6960Lm597mZBjD2IdL68+Dh/GO8pq8aV+OKUUM8SKkf0cbT33EvJoINbymzJIWiDWMFAOL4EnzDJzHMnxjJyKEXsQa+aTXqwRa9oR1cis+BLS8Aea0d0W82gg1jxxY2hRfIk3mjP9i/Lzd7dd/yGZYiEgbIXJhUuvoboYu6suc9Ed1mfubvsJKjVnJa0Hiurfq1Y9ltrYv4zkvuGZLE7OymPeGtxiHz/7D87a95+9G59uA/8Peu8ghBcVg8YAAAAASUVORK5CYII=)
Let E be the event that ‘the sum of numbers on the dice is 4’ and F be the event that ‘the two numbers appearing on throwing the two dice are different’.
⇒ E = {(1, 3), (2, 2), (3, 1)}
And ![](data:image/png;base64,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)
⇒ E ∩ F = {(1, 3), (3, 1)}
……….(i)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 15.Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.
Answer:The experiment is explained below in the tree diagram:
![](data:image/jpeg;base64,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)
The sample space of the given experiment is:
![](data:image/png;base64,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)
Let E be the event that ‘the coin shows a tail’ and F be the event that ‘at least one die shows a 3’.
⇒ E = {1T, 2T, 4T, 5T} and F = {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (6, 3)}
⇒ E ∩ F = ϕ ⇒ P(E ∩ F) = 0 ……….(i)
Now, we know that
By definition of conditional probability,
![](data:image/png;base64,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)
[Using (i)]
⇒ P(E|F) = 0
Question 16.If P (A) = 1/2, P(B) = 0, then P (A|B) is
A. 0
B. ![](data:image/png;base64,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)
C. not defined
D. 1
Answer:We know that
By definition of conditional probability,
……….(i)
Given: ![](data:image/png;base64,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)
And P(B) = 0
⇒ Using (i), we have
, which is not defined.
Question 17.If A and B are events such that P(A|B) = P(B|A), then
A. A ⊂ B but A ≠ B
B. A = B
C. A ∩ B = φ
D. P(A) = P(B)
Answer:Given: P(A|B) = P(B|A) ……….(i)
Now, we know that
By definition of conditional probability,
……….(ii)
……….(iii)
Using (i), we have
P(A|B) = P(B|A)
![](data:image/png;base64,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)
⇒ P(A) = P(B)
Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E)
Answer:
Given: P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2
We know that
By definition of conditional probability,
And
Question 2.
Compute P(A|B), if P(B) = 0.5 and P (A ∩ B) = 0.32
Answer:
Given: P(B) = 0.5 and P(A ∩ B) = 0.32
We know that
By definition of conditional probability,
Question 3.
If P (A) = 0.8, P (B) = 0.5 and P (B|A) = 0.4, find
(i) P(A ∩ B) (ii) P(A|B) (iii) P(A ∪ B)
Answer:
Given: P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4
(i) We know that
By definition of conditional probability,
⇒ P(A ∩ B) = P(B|A) P(A)
⇒ P(A ∩ B) = 0.4 × 0.8
⇒ P(A ∩ B) = 0.32
(ii) We know that
By definition of conditional probability,
⇒ P(A|B) = 0.64
(iii) Now, ∵ P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
⇒ P(A ∪ B) = 0.8 + 0.5 – 0.32 = 1.3 – 0.32
⇒ P(A ∪ B) = 0.98
Question 4.
Evaluate P(A ∪ B), if 2P(A) = P(B) = 5/13 and P(A|B) = 2/5.
Answer:
Given: and
……….(i)
We know that
By definition of conditional probability,
⇒ P(A ∩ B) = P(A|B) P(B)
……….(ii)
Now, ∵ P(A * B) = P(A) + P(B) – P(A ∩ B)
Question 5.
If P(A) = 6/11 , P(B) = 5/11 and P(A ∪ B) = 7/11, find
(i) P(A∩B) (ii) P(A|B) (iii) P(B|A)
Answer:
Given:
(i) We know that P(A * B) = P(A) + P(B) – P(A ∩ B)
⇒ P(A ∩ B) = P(A) + P(B) – P(A ∪ B)
(ii) Now, ∵ By definition of conditional probability,
(iii) Again, ∵ By definition of conditional probability,
Question 6.
A coin is tossed three times, where
(i) E : head on third toss, F : heads on first two tosses
(ii) E : at least two heads, F : at most two heads
(iii) E : at most two tails, F : at least one tail
Determine P(E|F)
Answer:
The sample space of the given experiment will be:
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
(i) Here, E: head on third toss
And F: heads on first two tosses
⇒ E = {HHH, HTH, THH, TTH} and F = {HHH, HHT}
⇒ E ∩ F = {HHH}
So,
Now, we know that
By definition of conditional probability,
(ii) Here, E: at least two heads
And F: at most two heads
⇒ E = {HHH, HHT, HTH, THH} and F = {HHT, HTH, THH, HTT, THT, TTH, TTT}
⇒ E ∩ F = {HHT, HTH, THH}
So,
Now, we know that
By definition of conditional probability,
(iii) Here, E: at most two tails
And F: at least one tail
⇒ E = {HHH, HHT, HTH, THH, HTT, THT, TTH}
And F = {HHT, HTH, THH, HTT, THT, TTH, TTT}
So,
Now, we know that
By definition of conditional probability,
Question 7.
Two coins are tossed once, where
(i) E : tail appears on one coin, F : one coin shows head
(ii) E : no tail appears, F : no head appears
Determine P(E|F)
Answer:
The sample space of the given experiment will be:
S = {HH, HT, TH, TT}
(i) Here, E: tail appears on one coin
And F: one coin shows head
⇒ E = {HT, TH} and F = {HT, TH}
⇒ E ∩ F = {HT, TH}
So,
Now, we know that
By definition of conditional probability,
⇒ P(E|F) = 1
(ii) Here, E: no tail appears
And F: no head appears
⇒ E = {HH} and F = {TT}
⇒ E ∩ F = ϕ
So,
Now, we know that
By definition of conditional probability,
⇒ P(E|F) = 0
Question 8.
A die is thrown three times,
E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses
Determine P(E|F)
Answer:
The sample space has 216 outcomes, where each element of the sample space has 3 entries and is of the form (x, y, z) where 1 ≤ x, y, z ≤ 6.
Here, E: 4 appears on the third toss
Now, F: 6 and 5 appears respectively on first two tosses
⇒ F = {(6, 5, 1), (6, 5, 2), (6, 5, 3), (6, 5, 4), (6, 5, 5), (6, 5, 6)}
⇒ E ∩ F = {(6, 5, 4)}
So,
Now, we know that
By definition of conditional probability,
Question 9.
Mother, father and son line up at random for a family picture
E : son on one end, F : father in middle
Determine P(E|F)
Answer:
Let M denote mother, F denote father and S denote son.
Then, the sample space for the given experiment will be:
S = {MFS, SFM, FSM, MSF, SMF, FMS}
Here, E: Son on one end
And F: Father in middle
⇒ E = {MFS, SFM, SMF, FMS} and F = {MFS, SFM}
⇒ E ∩ F = {MFS, SFM}
So,
Now, we know that
By definition of conditional probability,
⇒ P(E|F) = 1
Question 10.
A black and a red dice are rolled.
(a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.
(b) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.
Answer:
Let B denote black colored die and R denote red colored die.
Then, the sample space for the given experiment will be:
(a) Let A be the event of ‘obtaining a sum greater than 9’ and B be the event of ‘getting a 5 on black die’.
Then, A = {(B4, R6), (B5, R5), (B5, R6), (B6, R4), (B6, R5), (B6, R6)}
And B = {(B5, R1), (B5, R2), (B5, R3), (B5, R4), (B5, R5), (B5, R6)}
⇒ A ∩ B = {(B5, R5), (B5, R6)}
So,
Now, we know that
By definition of conditional probability,
(b) Let A be the event of ‘obtaining a sum 8’ and B be the event of ‘getting a number less than 4 on red die’.
Then, A = {(B2, R6), (B3, R5), (B4, R4), (B5, R3), (B6, R2)}
And
⇒ A ∩ B = {(B5, R3), (B6, R2)}
So,
Now, we know that
By definition of conditional probability,
Question 11.
A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5}
Find
(i) P(E|F) and P(F|E)
(ii) P(E|G) and P(G|E)
(iii) P((E ∪ F)|G) and P ((E ∩ F)|G)
Answer:
The sample space for the given experiment will be:
S = {1, 2, 3, 4, 5, 6}
Here, E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5} ……….(I)
……….(II)
Now, E ∩ F = {3}, F ∩ G = {2, 3}, E ∩ G = {3, 5} ……….(III)
……….(IV)
(i) We know that
By definition of conditional probability,
[Using (II) and (IV)]
Similarly, we have
[Using (II) and (IV)]
(ii) We know that
By definition of conditional probability,
Similarly, we have
(iii) Clearly, from (I), we have
E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5}
⇒ E ∪ F = {1, 2, 3, 5}
⇒ (E ∪ F) ∩ G = {2, 3, 5}
……….(V)
Now, we know that
By definition of conditional probability,
[Using (II) and (V)]
Similarly, we have E ∩ F = {3} [Using (III)]
And G = {2, 3, 4, 5} [Using (I)]
⇒ (E ∩ F) ∩ G = {3}
……….(VI)
So,
[Using (II) and (VI)]
Question 12.
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?
Answer:
Let B denote boy and G denote girl.
Then, the sample space of the given experiment will be:
S = {GG, GB, BG, BB}
Let E be the event that ‘both are girls’.
⇒ E = {GG}
(i) Let F be the event that ‘the youngest is a girl’.
⇒ F = {GG, BG}
……….(I)
Now, E ∩ F = {GG}
……….(II)
Now, we know that
By definition of conditional probability,
[Using (I) and (II)]
(ii) Let H be the event that ‘at least one is a girl’.
⇒ H = {GG, GB, BG}
……….(III)
Now, E ∩ H = {GG}
……….(IV)
Now, we know that
By definition of conditional probability,
[Using (III) and (IV)]
Question 13.
An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?
Answer:
Here, there are two types of questions, True/False or Multiple Choice Questions (T/F or MCQ), and each of them are divided into Easy and Difficult type, as shown below in the tree diagram.
So, in all, there are, 500 T/F questions and 900 MCQs.
Also, there are 800 Easy questions and 600 Difficult questions.
⇒ The sample space of this experiment has 500 + 900 = 1400 outcomes.
Now,
Let E be the event of ‘getting an Easy question’ and F be the event of ‘getting an MCQ’.
and
……….(i)
Now, E ∩ F is the event of getting an MCQ which is Easy.
Clearly, from the diagram, we know that there are 500 MCQs that are easy.
So, ……….(ii)
Now, we know that
By definition of conditional probability,
[Using (i) and (ii)]
Question 14.
Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4’.
Answer:
The sample space of the given experiment is:
Let E be the event that ‘the sum of numbers on the dice is 4’ and F be the event that ‘the two numbers appearing on throwing the two dice are different’.
⇒ E = {(1, 3), (2, 2), (3, 1)}
And
⇒ E ∩ F = {(1, 3), (3, 1)}
……….(i)
Now, we know that
By definition of conditional probability,
Question 15.
Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.
Answer:
The experiment is explained below in the tree diagram:
The sample space of the given experiment is:
Let E be the event that ‘the coin shows a tail’ and F be the event that ‘at least one die shows a 3’.
⇒ E = {1T, 2T, 4T, 5T} and F = {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (6, 3)}
⇒ E ∩ F = ϕ ⇒ P(E ∩ F) = 0 ……….(i)
Now, we know that
By definition of conditional probability,
[Using (i)]
⇒ P(E|F) = 0
Question 16.
If P (A) = 1/2, P(B) = 0, then P (A|B) is
A. 0
B.
C. not defined
D. 1
Answer:
We know that
By definition of conditional probability,
……….(i)
Given:
And P(B) = 0
⇒ Using (i), we have
, which is not defined.
Question 17.
If A and B are events such that P(A|B) = P(B|A), then
A. A ⊂ B but A ≠ B
B. A = B
C. A ∩ B = φ
D. P(A) = P(B)
Answer:
Given: P(A|B) = P(B|A) ……….(i)
Now, we know that
By definition of conditional probability,
……….(ii)
……….(iii)
Using (i), we have
P(A|B) = P(B|A)
⇒ P(A) = P(B)
Exercise 13.2
Question 1.If P(A) = 3/5 and P (B) = 1/5, find P (A ∩ B) if A and B are independent events.
Answer:Given:![](data:image/png;base64,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)
As A and B are independent events.
⇒ P (A ∩ B) = P(A).P(B)
![](data:image/png;base64,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)
Question 2.Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
Answer:Given: A pack of 52 cards.
As we know there are 26 cards in total which are black. Let A and B denotes respectively the events that the first and second drawn cards are black.
Now, P(A) = P(black card in first draw) = ![](data:image/png;base64,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)
Because the second card is drawn without replacement so, now the total number of black card will be 25 and total cards will be 51. i.e. the conditional probability of B given that A has already occurred.
Now, P
= P(black card in second draw) = ![](data:image/png;base64,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)
Thus the probability that both the cards are black:
⇒ P(A ∩ B) =![](data:image/png;base64,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)
Hence, the probability that both the cards are black =
.
Question 3.A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
Answer:Given: A box of oranges.
Let A, B and C denotes respectively the events that the first, second and third drawn orange is good.
Now, P(A) = P(good orange in first draw) = ![](data:image/png;base64,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)
Because the second orange is drawn without replacement so, now the total number of good oranges will be 11 and total oranges will be 14. i.e. the conditional probability of B given that A has already occurred.
Now, P
= P(good orange in second draw) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAgCAMAAAD68tKbAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAABmADqQAGaQAGa2OgAAOjoAOmZmOma2OpDbZjoAZjpmZrbbZrb/kDoAkDo6kGaQtmYAtmY6tv//25A625Bm29uQ2////7Zm/9uQ//+2///bTz76BgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAcElEQVQoU52Qxw7AIAxD0z0pXXTm/7+zJRTIDVFfLCtOFD2AoHApqPP5VneUrQMoM//rOOcH3Teuklet9+B/cQV9nSlu+Vfb8oOrHDg3HCudHTclJM9ngzzf/YpSTK5PmJLMc3sBUt/yA9jLdA19/AA+IgXLwlamiAAAAABJRU5ErkJggg==)
Because the third orange is drawn without replacement so, now the total number of good oranges will be 10 and total orangs will be 13. i.e. the conditional probability of C given that A nd B has already occurred.
Now, P
= P(good orange in third draw) = ![](data:image/png;base64,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)
Thus the probability that all the oranges are good:
⇒ P(A ∩ B ∩ C) =![](data:image/png;base64,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)
Hence, the probability that a box will be approved for sale ![](data:image/png;base64,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)
Question 4.A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.
Answer:Given: A fair coin and an unbiased die are tossed.
We know that the sample space S:
S = {(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}
Let A be the event ‘head appears on the coin:
⇒ A = {(H,1), (H,2), (H,3), (H,4), (H,5), (H,6)}
⇒ P(A) = ![](data:image/png;base64,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)
Now, Let B be the event 3 on the die:
⇒ B = {(H,3), (T,3)}
![](data:image/png;base64,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)
As, A ∩ B = {(H,3)}
⇒ P(A ∩ B) =
..(1)
And P(A) . P(B) =
..(2)
From (1) and (2) P (A ∩ B) = P(A) . P(B)
Therefore, A and B are independent events.
Question 5.A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, ‘the number is even,’ and B be the event, ‘the number is red’. Are A and B independent?
Answer:Given: A die is given.
The sample space for the dice will be:
S = {1, 2, 3, 4, 5, 6}
Let A be the event, the number is even:
⇒ A = {2, 4, 6}
![](data:image/png;base64,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)
Now, Let B be the event, the number is red:
⇒ B = {1, 2, 3}
⇒ P(B) = ![](data:image/png;base64,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)
As, A ∩ B = {2}
⇒ P(A ∩ B) =
..(1)
And P(A) . P(B) =
..(2)
From (1) and (2) P (A ∩ B) ≠ P(A) . P(B)
Therefore, A and B are not independent events.
Question 6.Let E and F be events with P (E)= 3/5, P (F) = 3/10 and P (E ∩ F) = 1/5. Are E and F independent?
Answer:Given: P(E) =
, P(F) =
and P(E ∩ F) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAEJQTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOma2OpDbZjoAZrbbZrb/kDoAkDo6kNv/tmYAtmY625A625Bm2////9uQ///b7M5aaQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAU0lEQVQYV2NgwACiAiwgMSE2TjDNwCCInxblZxYGqWIEAg5M43CIgFQDAdHq4QpB9jDxAe3jhroORgPF2SFCIqwQMVEeDgZRXj4wX1QAqI0LwzoABl4CTEIhdkMAAAAASUVORK5CYII=)
P(E) . P(F) = ![](data:image/png;base64,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)
⇒ P (E ∩ F) ≠ P(E) . P(F)
Therefore, E and F are not independent events.
Question 7.Given that the events A and B are such that P(A) = 1/2, P (A ∪ B) = 3/5 and P(B) = p. Find p if they are (i) mutually exclusive (ii) independent.
Answer:Given: P(A) =
, P (A ∪ B) =
and P(B) = p
(i) mutually exclusive
When A and B are mutually exclusive.
Then (A ∩ B) = ϕ
⇒ P (A ∩ B) = 0
As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
(ii) independent
When A and B are independent.
⇒ P (A ∩ B) = P(A) . P(B)
⇒ P (A ∩ B) = ![](data:image/png;base64,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)
As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Question 8.Let A and B be independent events with P (A) = 0.3 and P(B) = 0.4. Find
(i) P(A ∩ B) (ii) P(A ∪ B)
(iii) P (A|B) (iv) P (B|A)
Answer:Given: P(A) = 0.3 and P(B) = 0.4
(i) P(A ∩ B)
When A and B are independent.
⇒ P (A ∩ B) = P(A) . P(B)
⇒ P (A ∩ B) = 0.3 × 0.4
⇒ P (A ∩ B) = 0.12
(ii) P(A ∪ B)
As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒ P (A ∪ B) = 0.3 + 0.4 – 0.12
⇒ P (A ∪ B) = 0.58
(iii) P (A|B)
As we know ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIYAAAAkCAMAAAB7c7JGAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZmaQZma2ZpCQZpDbZrbbZrb/kDoAkDo6kDpmkGYAkNv/tmYAtmY6tpA6tpBmttu2ttv/tv/btv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2/9vb//+2///bXpw1ZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACxUlEQVRYR+1XW3vTMAx1WjoCYyvLuAdYBhSydc6I//+PQ9KxXGe51ECyBz700CWZIh9LR/KJMf9twQw0eZZlq3e6wv71zpi2eNpfURyzze7mzd0ScOz6h7nNPiB0fcG/9pSewW7zbPUel+x4TwCbsyVwcHRXyvJ+BffpW+Fh2fXO7HPcsKOpTu481plTEsPwydiaClVxJSOoccOOzSU9aJ5rrmaEwtFvVleHVc31FXZOJLnk574KlrlxQQVxH8V7XuPoG8TF5sFFKUT7itEoDOZGSZmB28zmNx7VgHjieLkBGKYmbgzDcF/POrVSvinc+px6cNzsOqRY4t8LPa+lTCPZoKK4klOWvQihXfkSDeRKQip/ZTvittrSo+Z0IolUgtCd3AP0IiGgSnGwGAZqdc7xmKIyW/a5vnvYu+X3FQbc3JeMk9s8S+RUWg9IP6FstQ6cQ2mrt7l0P4DBrS0kQ1zRJKu3x93AWMS3GQaO55LU8Tuq0oUhVEtOh5FhPml+mHez0fgRJ0OmlkpFMH5+3gh7H7L2+JaPegiMfe7TbMEHNDMwKQxm1Infm5+KNIyFucLeRL6MAJIekA5gCzC4YkhUlA1X+cX4KBi2AOt3Ljps4LioA19IGL4boKicSPPagZRxNjD9LfdPDMOVgBlgzFiUWJNoUaxsF8Oi37B6Ys6YkK408p2iZ0ARxAMQVejrtkB7z2fM0KjQOqXkYVswO+Jh/gTKKX1u/CnQ6ID0IQaGROoUjYQpzTFVnyZBfvYX7T9JT0YQpnJuqPpMkZ/hhNV89mBMnrDdKqgixLkR1GeK/PxLvdHBoTC8KFX1uYz8HOehF6ZeXwX1uYz8nIABYaowVH0uIz+ns0H/jWBAqyyigqdgxF8K0ik4Lpf4JhiFEYSpUPSgPh+bogrwoShF5zy+dUXpMl/PKbuKRWnCME8J+c/6/ALp3VZvJE2F+AAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒ P (A|B) = 0.3
(iv) P (B|A)
As we know ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIYAAAAkCAMAAAB7c7JGAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZmaQZma2ZpCQZpDbZrbbZrb/kDoAkDo6kDpmkGYAkNv/tmYAtmY6tpA6tpBmttu2ttv/tv/btv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2/9vb//+2///bXpw1ZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACz0lEQVRYR+1Xa1PbMBCUkgbcUkgx9Om2mNJQQ5Cp9f9/XO+pyIntqKmdmc6gD8TWyHer29VpMeZlTFiBOrPWzj5phvX7lTFNfrqbkRbaxerhw9MUcNz8l3m0Xzh0dYl/3RnM8XjM7OwzP+LCZwBYn0+BA6P7gtJLBv/tZy6w3Hxl1hm/4EJTnjwJ1pFLEsOQYixNyaz4AhFU/IIL6yuYqN9orUaEgtEfZjebrOb2hncOIrnCeWHBoTYugRD/lVaPOzD6guPy5lmLRERzjWgUBmqjgMrwspGHbDziAHTiMV0HDFOBNrph+LvzFleqN4VbXcAZ7B9uHkpM8Z9JnrdEU081gBRfYMns2xDaF+8QIM3OlkidyB4eADoW9WygiEBBOJ14BiASIACm8NsYBnN1AbMkUeot60y/laQ0639Yoo5PH8Ri5Zn6daKm0s4AnSemrdKGI9TybJMTdQKj/JjJEzKaNKrl/mWsWE7oLGcQLckv10RgNNf3wkp6OQw188EhzbxdjVpaHM3+/r4A1hQGdJ1KqNtW7f4t711BCdeZlNkJ/azRE9yK5ETZK0jtitCMScok50S99ACihHQkSIYBBtJR4pvAQAqVMroKukeA9TcPkRo46qbs2xKtKK6w0g9jb/V7qxGbgVY1YPfhuuTrwMmBCjBGJKUfRnRgHdGgBkZuzEO33vFd2xqpCPmolniMSRt6KeTESpNLAxkNByo00pukk2b+Co0SwmhyWoU/iCO5jR4MM7ogJUZHk0jtopEx5T7GpU+wn7tJd2fSixGMKfswdaUJ9pNu2HjswBi8Yds0qCOUvMGVCqohzv7Rb7RCKwwthrrStKv3YGVtfyjGVP1VcKWT2M9+1GJMBcbGlU5iPwdgcPOXrEgNd4Sjw4j+U4hc6XFJCcaU67BxpUeWqNK1nTbhwI52TOJAbVOa0L4mQQH+LjKlCc18IhT/R9g/iNZVwbeBwrsAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒ P (B|A) = 0.4
Question 9.If A and B are two events such that P (A) = 1/4 , P (B) = 1/2 and P (A ∩ B) = 1/8, find P (not A and not B).
Answer:Given: P (A) =
, P (B) =
and P (A ∩ B) =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAA1CAYAAACnUADaAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAF9SURBVEjH7ZYxbsJAEEUnPRwAiXFHelifwVopUFLYFq0Ly3JDJBdRtLhzwwHooMwF0kCZKicgEjlBOAPOjg3IkTb2mlRWjDQNkp9n5v/9a4jjGP5a0ELqQ+Tv7maI709HlsWeEeEDrWBeG7Jw2RgMY28YsJedpDdBfsBaSAtpBIQOnrDZhCDAnE3ZQVT+KUI+RIAv+eCpWL911Mbj/4KQoXSrDHLSrVadxkOqvFEJiSKv6zBcFoMJmbP0oqirBUkSr8P7sAPkWzsUgzzp7AFH2EKf77wk6VRC8nDGIw/FUBGZR+YuxpWQwMK5EiJmPQbwqQW5vJESvriDrEM5otY4VC6DdXZV4Oh96j/dEwDRfL3sSFtiUicD0eVVokwpJBsL8e0sc1pL4vNyHxH5C81/lZe6qicxpEUVyDuXPaluQcUuYKOS+GpCqZqmT2QntpioIe662ie5qQ7UjWUHD3Tw6Cy5HAUAO8yE6Gl+FchlmsaqGMyGyVe1fdLMZPsGbwQEL3Rp/KYAAAAASUVORK5CYII=)
P(not A and not B) = P(A’ ∩ B’)
As, { A’ ∩ B’ =(A ∪ B)’}
⇒ P(not A and not B) = P ((A ∪ B)’)
= 1 - P (A ∪ B)
= 1- [P(A) + P(B) - P (A ∩ B)]
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 10.Events A and B are such that P (A) = 1/2, P(B) = 7/12 and P(not A or not B) = 1/4. State whether A and B are independent ?
Answer:Given: P (A) =
, P(B) =
and P(not A or not B) = 1/4
⇒ P (A’∪ B’) = 1/4
⇒ P (A ∩ B)’ = 1/4
⇒ 1 - P (A ∩ B) = 1/4
⇒ P (A ∩ B) = 1 - 1/4
⇒ P (A ∩ B) = 3/4 ..(1)
And P(A) . P(B) =
..(2)
From (1) and (2) P (A ∩ B) ≠ P(A) . P(B)
Therefore, A and B are not independent events.
Question 11.Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6.
Find
(i) P(A and B) (ii) P(A and not B)
(iii) P(A or B) (iv) P(neither A nor B)
Answer:Given: P(A) = 0.3, P(B) = 0.6.
(i) P(A and B)
As A and B are independent events.
⇒ P(A and B) = P (A ∩ B) = P(A) . P(B)
= 0.3 × 0.6
= 0.18
(ii) P(A and not B)
⇒ P(A and not B) = P (A ∩ B’) = P(A) - P(A ∩ B)
= 0.3 - 0.18
= 0.12
(iii) P(A or B)
⇒ P(A or B) = P(A ∪ B)
As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒ P (A ∪ B) = 0.3 + 0.6 – 0.18
⇒ P (A ∪ B) = 0.72
(iv) P(neither A nor B)
P(neither A nor B) = P(A’ ∩ B’)
As, { A’ ∩ B’ =(A ∪ B)’}
⇒ P(neither A nor B) = P ((A ∪ B)’)
= 1 - P (A ∪ B)
= 1 - 0.72
= 0.28
Question 12.A die is tossed thrice. Find the probability of getting an odd number at least once.
Answer:Given: A die is tossed thrice.
The sample space S = {1, 2, 3, 4, 5, 6}
Let P(A) = probability of getting an odd number in first throw.
⇒ P(A) =
.
Let P(B) = probability of getting an even number.
⇒ P(B) =
.
Now, probability of getting an even number in three times = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAF4AAAAgCAMAAACsJD/tAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6ADpmADqQAGaQAGa2OgAAOgA6OmZmOmaQOma2OpDbZgAAZgA6ZjoAZjpmZmaQZrbbZrb/kDoAkLbbkNv/tmYAtmY6tv//25A625Bm27Zm29u22////7Zm/9uQ/9u2//+2///b2j372AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABSElEQVRIS92VUVPCMAyAM0VQxpxgZTphsPb//0fTtLvZ0tReOB6wL7vjW742WVIA7nyZzyWbgZSh0Id+v7SsXsrQPof2/OlBynCDKVSqyMX9F73pngaudaTMdo7T9hWuTdovZX9obzPJ+u2LxKN/BrtkYdF5jHq0ft3UidezsEgPRi0GtKfHLAuL/cd2yTQS+jnY2f6g9bAPdvK/Tr8ZVWECfkUMk5thzKLjMzjQxxlnYUl5KP/5+GFIBrLFCQzT10sWPwsLz27F6dYhOwdL7DBu3bHP/hkEZWGR/hYvHduqSk2p3UvK7MWC2jXO6OsHHKJZmLKQMow3agNj4y7i83M4ar+LJGS6eQfo3O3SM9f9FcyoehjtFgAHrvRXMKp9TX9XGbuUgW53WH8szgntJ8rickkZfs6V0+vGXm5pvZRRxVfUmHe9fgAFrRq0mSA3uQAAAABJRU5ErkJggg==)
So, probability of getting an odd number at least once
= 1 – probability of getting an odd number in no throw
= 1 - probability of getting an even number in three times
![](data:image/png;base64,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)
∴ probability of getting an odd number at least once = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAEVQTFRFAAAAAAAAAGa2OgAAOmaQOma2OpDbZgAAZjoAZrbbZrb/kDoAkDo6kDqQkNv/tmYA27Zm2////7Zm/9uQ/9u2//+2///bh/RrqQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXUlEQVQYV41OWw6AIAwrIA+diihw/6M6ZmIMicb+dF33KNChkmJo1CWhzqu4uxW6pf6S/cEX3b4wfk4/xnJQauBs5JC9Q/ETEDkv2ZRb3XybUMLIPY3DXIzNyF6HE4LTA1nCRZk+AAAAAElFTkSuQmCC)
Question 13.Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that
(i) both balls are red.
(ii) first ball is black and second is red.
(iii) one of them is black and other is red.
Answer:Given: A box containing 10 black and 8 red balls.
Total number of balls in box = 18
(i) both balls are red.
Probability of getting a red ball in first draw = ![](data:image/png;base64,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)
As the ball is replaced after first throw,
Hence, Probability of getting a red ball in second draw = ![](data:image/png;base64,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)
Now, Probability of getting both balls red = ![](data:image/png;base64,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)
(ii) first ball is black and second is red.
Probability of getting a black ball in first draw = ![](data:image/png;base64,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)
As the ball is replaced after first throw,
Hence, Probability of getting a red ball in second draw = ![](data:image/png;base64,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)
Now, Probability of getting first ball is black and second is red = ![](data:image/png;base64,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)
(iii) one of them is black and other is red.
Probability of getting a black ball in first draw = ![](data:image/png;base64,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)
As the ball is replaced after first throw,
Hence, Probability of getting a red ball in second draw = ![](data:image/png;base64,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)
Now, Probability of getting first ball is black and second is red = ![](data:image/png;base64,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)
Probability of getting a red ball in first draw = ![](data:image/png;base64,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)
As the ball is replaced after first throw,
Hence, Probability of getting a black ball in second draw = ![](data:image/png;base64,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)
Now, Probability of getting first ball is red and second is black = ![](data:image/png;base64,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)
Therefore, Probability of getting one of them is black and other is red :
= Probability of getting first ball is black and second is red + Probability of getting first ball is red and second is black
= ![](data:image/png;base64,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)
Question 14.Probability of solving specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved (ii) exactly one of them solves the problem.
Answer:Given:
P(A) = Probability of solving the problem by A = 1/2
P(B) = Probability of solving the problem by B = 1/3
Because A and B both are independent.
⇒ P (A ∩ B) = P(A) . P(B)
⇒ P (A ∩ B) = ![](data:image/png;base64,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)
P(A’) = 1 – P(A) = 1 – 1/2 = 1/2
P(B’) = 1 – P(B) = ![](data:image/png;base64,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)
(i) the problem is solved
The problem is solved, i.e. it is either solved by A or it is solved by B.
= P(A ∪ B)
As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒ P (A ∪ B) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
(ii) exactly one of them solves the problem
i.e. either problem is solved by A but not by B or vice versa
i.e. P(A).P(B’) + P(A’).P(B)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ P(A).P(B’) + P(A’).P(B) = 1/2
Question 15.One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent?
(i) E : ‘the card drawn is a spade’
F : ‘the card drawn is an ace’
(ii) E : ‘the card drawn is black’
F : ‘the card drawn is a king’
(iii) E : ‘the card drawn is a king or queen’
F : ‘the card drawn is a queen or jack’.
Answer:Given: A deck of 52 cards.
(i) In a deck of 52 cards, 13 cards are spade and 4 cards are ace and only one card is there which is spade and ace both.
Hence, P(E) = The card drawn is a spade = ![](data:image/png;base64,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)
P(F) = The card drawn is an ace = ![](data:image/png;base64,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)
P(E ∩ F) = The card drawn is a spade and ace both =
..(1)
And P(E) . P(F)
..(2)
From (1) and (2)
⇒ P (E ∩ F) = P(E) . P(F)
Hence, E and F are independent events.
(ii) In a deck of 52 cards, 26 cards are black and 4 cards are king and only two card are there which are black and king both.
Hence, P(E) = The card drawn is of black = ![](data:image/png;base64,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)
P(F) = The card drawn is a king = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADUAAAAgCAMAAACFI8BqAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OjoAOmZmOma2OpDbZgAAZgA6ZjoAZjpmZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkNv/tmYAtmY6tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///b8LuJ4wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA9klEQVQ4T91U2Q6CMBBs8UJEQRFUjoIHyP7/D9oLTBNKgQcS3acmnenMbttBaP6q7LNZFNKNAoLIMbPyna+ySBCaWQgRhVW6MJ5VHzMIg+vL2JmiRTCr1UgW1RjkEJK1evLDXmQmg9zS3oT6v/2EXwUrK+7srt0XC80EhqEmjc/ocNKpv0Bib5LdytPH2NUblpmRO9g6URQRX7/2bqjQ3CndlpkBkURJFmNW2+6XwI9t/zFHMYfSGen7PA3rfQlEHzIMi562Wq0ELyULQiZCeklfh3cauBDFQqukpFIfiiIzak+gIaWDp5r1gb1rLavJjNwWk5+vPu46Dq5BRWBqAAAAAElFTkSuQmCC)
P(E ∩ F) = The card drawn is a black and king both =
..(1)
And P(E) . P(F)
..(2)
From (1) and (2)
⇒ P (E ∩ F) = P(E) . P(F)
Hence, E and F are independent events.
(iii) In a deck of 52 cards, 4 cards are queen, 4 cards are king and 4 cards are jack.
Hence, P(E) = The card drawn is either king or queen = ![](data:image/png;base64,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)
P(F) = The card drawn is either queen or jack = ![](data:image/png;base64,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)
There are 4 cards which are either king or queen and either queen or jack.
P(E ∩ F) = The card drawn is either king or queen and either queen or jack =
..(1)
And P(E) . P(F)
..(2)
From (1) and (2)
⇒ P (E ∩ F) ≠ P(E) . P(F)
Hence, E and F are not independent events.
Question 16.In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random.
(a) Find the probability that she reads neither Hindi nor English newspapers.
(b) If she reads Hindi newspaper, find the probability that she reads English newspaper.
(c) If she reads English newspaper, find the probability that she reads Hindi newspaper
Answer:Given:
Let H and E denote the number of students who read Hindi and English newspaper respectively.
Hence, P(H) = Probability of students who read Hindi newspaper= ![](data:image/png;base64,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)
P(E) = Probability of students who read English newspaper = ![](data:image/png;base64,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)
P (H ∩ E) = Probability of students who read Hindi and English both newspaper = ![](data:image/png;base64,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)
(a) Find the probability that she reads neither Hindi nor English newspapers.
P(neither H nor E)
P(neither H nor E) = P(H’ ∩ E’)
As, { H’ ∩ E’ =(H ∪ E)’}
⇒ P(neither A nor B) = P ((H ∪ E)’)
= 1 - P (H ∪ E)
= 1- [P(H) + P(E) - P (H ∩ E)]
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(b) If she reads Hindi newspaper, find the probability that she reads English newspaper.
P (E|H) = hindi newspaper reading has already occurred and the probability that she reads English newspaper is to find.
As we know ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ P (E|H) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOpDbZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ2////9uQ/9u2//+2///bjLFCZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAW0lEQVQYV42MSQ6AIBAEG1TcV1BUlP8/U2YOHiQa61JJp2aACD9ntNmyYQPLt71Jd6pEoIrfvSxUB373d2gLIVvADxNWqXl2OfsYO5IRCRvYVI+z1nDBsIrvHlxvGQNJwgEIiAAAAABJRU5ErkJggg==)
(c) If she reads English newspaper, find the probability that she reads Hindi newspaper.
P (H|E) = English newspaper reading has already occurred and the probability that she reads Hindi newspaper is to find.
As we know ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ P (H|E) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAWUlEQVQYV2NgwACSAiwgMSE2bjDNwCCIn5bkZxYFqWIEAg5M43CIgFQDAdHq4QpFuBkZ2RkYJLh4GYSZ+MDC4qwQWhBivTBQGsSDUGJASoyHQYITZB0Pun0AG8sCbODE2DAAAAAASUVORK5CYII=)
Question 17.The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
A. 0
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:Given: A pair of dice is rolled.
Hence the number of outcomes = 36
Let P(E) be the probability to get an even prime number on each die.
As we know the only even prime number is 2.
So, E = {2,2}
⇒ ![](data:image/png;base64,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)
The correct answer is D.
Question 18.Two events A and B will be independent, if
(A) A and B are mutually exclusive
(B) P(A′B′) = [1 – P(A)] [1 – P(B)]
(C) P(A) = P(B)
(D) P(A) + P(B) = 1
Answer:Given: Two events A and B will be independent.
As A and B are independent events.
⇒ P (A ∩ B) = P(A) . P(B)
We solve it using options.
Let P(A) = a, P(B) = b
As, A and B are mutually exclusive
P (A ∩ B) = ϕ
Now, P(A).P(B) = a.b ≠ P(A ∩ B)
⇒ P (A ∩ B) ≠ P(A) . P(B)
Hence, it shows A and B are not Independent events.
(B) P(A′B′) = [1 – P(A)] [1 – P(B)]
⇒ P(A′∩ B′) = 1 – P(A) – P(B) + P(A)P(B)
⇒ 1 - P (A ∪ B) =1 – P(A) – P(B) + P(A)P(B)
= - [P(A) + P(B) - P (A ∩ B)] = – P(A) – P(B) + P(A)P(B)
= - P(A) - P(B) + P (A ∩ B) = – P(A) – P(B) + P(A)P(B)
⇒ P (A ∩ B) = P(A) . P(B)
Hence, it shows A and B are Independent events.
(C) P(A) = P(B)
As, P(A) = P(B)
Let we take the example of a coin
P(A) = probability of getting head = 1/2
P(B) = probability of getting tail = 1/2
A ∩ B = ϕ
P(A ∩ B) = probability of getting head and tail both = 0
Now, P(A).P(B) = 1/2 . 1/2 = 1/4 ≠ P(A ∩ B)
⇒ P (A ∩ B) ≠ P(A) . P(B)
Hence, it shows A and B are not Independent events.
(D) P(A) + P(B) = 1
Let we take the example of a coin
P(A) = probability of getting head = 1/2
P(B) = probability of getting tail = 1/2
Now, P(A) + P(B) = 1/2 + 1/2 = 1
But it doesnot inferred that A and B are independent.
Hence the correct option is B.
If P(A) = 3/5 and P (B) = 1/5, find P (A ∩ B) if A and B are independent events.
Answer:
Given:
As A and B are independent events.
⇒ P (A ∩ B) = P(A).P(B)
Question 2.
Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
Answer:
Given: A pack of 52 cards.
As we know there are 26 cards in total which are black. Let A and B denotes respectively the events that the first and second drawn cards are black.
Now, P(A) = P(black card in first draw) =
Because the second card is drawn without replacement so, now the total number of black card will be 25 and total cards will be 51. i.e. the conditional probability of B given that A has already occurred.
Now, P = P(black card in second draw) =
Thus the probability that both the cards are black:
⇒ P(A ∩ B) =
Hence, the probability that both the cards are black = .
Question 3.
A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
Answer:
Given: A box of oranges.
Let A, B and C denotes respectively the events that the first, second and third drawn orange is good.
Now, P(A) = P(good orange in first draw) =
Because the second orange is drawn without replacement so, now the total number of good oranges will be 11 and total oranges will be 14. i.e. the conditional probability of B given that A has already occurred.
Now, P = P(good orange in second draw) =
Because the third orange is drawn without replacement so, now the total number of good oranges will be 10 and total orangs will be 13. i.e. the conditional probability of C given that A nd B has already occurred.
Now, P= P(good orange in third draw) =
Thus the probability that all the oranges are good:
⇒ P(A ∩ B ∩ C) =
Hence, the probability that a box will be approved for sale
Question 4.
A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.
Answer:
Given: A fair coin and an unbiased die are tossed.
We know that the sample space S:
S = {(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}
Let A be the event ‘head appears on the coin:
⇒ A = {(H,1), (H,2), (H,3), (H,4), (H,5), (H,6)}
⇒ P(A) =
Now, Let B be the event 3 on the die:
⇒ B = {(H,3), (T,3)}
As, A ∩ B = {(H,3)}
⇒ P(A ∩ B) = ..(1)
And P(A) . P(B) = ..(2)
From (1) and (2) P (A ∩ B) = P(A) . P(B)
Therefore, A and B are independent events.
Question 5.
A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, ‘the number is even,’ and B be the event, ‘the number is red’. Are A and B independent?
Answer:
Given: A die is given.
The sample space for the dice will be:
S = {1, 2, 3, 4, 5, 6}
Let A be the event, the number is even:
⇒ A = {2, 4, 6}
Now, Let B be the event, the number is red:
⇒ B = {1, 2, 3}
⇒ P(B) =
As, A ∩ B = {2}
⇒ P(A ∩ B) = ..(1)
And P(A) . P(B) = ..(2)
From (1) and (2) P (A ∩ B) ≠ P(A) . P(B)
Therefore, A and B are not independent events.
Question 6.
Let E and F be events with P (E)= 3/5, P (F) = 3/10 and P (E ∩ F) = 1/5. Are E and F independent?
Answer:
Given: P(E) = , P(F) =
and P(E ∩ F) =
P(E) . P(F) =
⇒ P (E ∩ F) ≠ P(E) . P(F)
Therefore, E and F are not independent events.
Question 7.
Given that the events A and B are such that P(A) = 1/2, P (A ∪ B) = 3/5 and P(B) = p. Find p if they are (i) mutually exclusive (ii) independent.
Answer:
Given: P(A) =, P (A ∪ B) =
and P(B) = p
(i) mutually exclusive
When A and B are mutually exclusive.
Then (A ∩ B) = ϕ
⇒ P (A ∩ B) = 0
As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒
⇒
(ii) independent
When A and B are independent.
⇒ P (A ∩ B) = P(A) . P(B)
⇒ P (A ∩ B) =
As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒
⇒
⇒
Question 8.
Let A and B be independent events with P (A) = 0.3 and P(B) = 0.4. Find
(i) P(A ∩ B) (ii) P(A ∪ B)
(iii) P (A|B) (iv) P (B|A)
Answer:
Given: P(A) = 0.3 and P(B) = 0.4
(i) P(A ∩ B)
When A and B are independent.
⇒ P (A ∩ B) = P(A) . P(B)
⇒ P (A ∩ B) = 0.3 × 0.4
⇒ P (A ∩ B) = 0.12
(ii) P(A ∪ B)
As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒ P (A ∪ B) = 0.3 + 0.4 – 0.12
⇒ P (A ∪ B) = 0.58
(iii) P (A|B)
As we know
⇒
⇒ P (A|B) = 0.3
(iv) P (B|A)
As we know
⇒
⇒ P (B|A) = 0.4
Question 9.
If A and B are two events such that P (A) = 1/4 , P (B) = 1/2 and P (A ∩ B) = 1/8, find P (not A and not B).
Answer:
Given: P (A) =, P (B) =
and P (A ∩ B) =
P(not A and not B) = P(A’ ∩ B’)
As, { A’ ∩ B’ =(A ∪ B)’}
⇒ P(not A and not B) = P ((A ∪ B)’)
= 1 - P (A ∪ B)
= 1- [P(A) + P(B) - P (A ∩ B)]
Question 10.
Events A and B are such that P (A) = 1/2, P(B) = 7/12 and P(not A or not B) = 1/4. State whether A and B are independent ?
Answer:
Given: P (A) =, P(B) =
and P(not A or not B) = 1/4
⇒ P (A’∪ B’) = 1/4
⇒ P (A ∩ B)’ = 1/4
⇒ 1 - P (A ∩ B) = 1/4
⇒ P (A ∩ B) = 1 - 1/4
⇒ P (A ∩ B) = 3/4 ..(1)
And P(A) . P(B) = ..(2)
From (1) and (2) P (A ∩ B) ≠ P(A) . P(B)
Therefore, A and B are not independent events.
Question 11.
Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6.
Find
(i) P(A and B) (ii) P(A and not B)
(iii) P(A or B) (iv) P(neither A nor B)
Answer:
Given: P(A) = 0.3, P(B) = 0.6.
(i) P(A and B)
As A and B are independent events.
⇒ P(A and B) = P (A ∩ B) = P(A) . P(B)
= 0.3 × 0.6
= 0.18
(ii) P(A and not B)
⇒ P(A and not B) = P (A ∩ B’) = P(A) - P(A ∩ B)
= 0.3 - 0.18
= 0.12
(iii) P(A or B)
⇒ P(A or B) = P(A ∪ B)
As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒ P (A ∪ B) = 0.3 + 0.6 – 0.18
⇒ P (A ∪ B) = 0.72
(iv) P(neither A nor B)
P(neither A nor B) = P(A’ ∩ B’)
As, { A’ ∩ B’ =(A ∪ B)’}
⇒ P(neither A nor B) = P ((A ∪ B)’)
= 1 - P (A ∪ B)
= 1 - 0.72
= 0.28
Question 12.
A die is tossed thrice. Find the probability of getting an odd number at least once.
Answer:
Given: A die is tossed thrice.
The sample space S = {1, 2, 3, 4, 5, 6}
Let P(A) = probability of getting an odd number in first throw.
⇒ P(A) = .
Let P(B) = probability of getting an even number.
⇒ P(B) = .
Now, probability of getting an even number in three times =
So, probability of getting an odd number at least once
= 1 – probability of getting an odd number in no throw
= 1 - probability of getting an even number in three times
∴ probability of getting an odd number at least once =
Question 13.
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that
(i) both balls are red.
(ii) first ball is black and second is red.
(iii) one of them is black and other is red.
Answer:
Given: A box containing 10 black and 8 red balls.
Total number of balls in box = 18
(i) both balls are red.
Probability of getting a red ball in first draw =
As the ball is replaced after first throw,
Hence, Probability of getting a red ball in second draw =
Now, Probability of getting both balls red =
(ii) first ball is black and second is red.
Probability of getting a black ball in first draw =
As the ball is replaced after first throw,
Hence, Probability of getting a red ball in second draw =
Now, Probability of getting first ball is black and second is red =
(iii) one of them is black and other is red.
Probability of getting a black ball in first draw =
As the ball is replaced after first throw,
Hence, Probability of getting a red ball in second draw =
Now, Probability of getting first ball is black and second is red =
Probability of getting a red ball in first draw =
As the ball is replaced after first throw,
Hence, Probability of getting a black ball in second draw =
Now, Probability of getting first ball is red and second is black =
Therefore, Probability of getting one of them is black and other is red :
= Probability of getting first ball is black and second is red + Probability of getting first ball is red and second is black
=
Question 14.
Probability of solving specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved (ii) exactly one of them solves the problem.
Answer:
Given:
P(A) = Probability of solving the problem by A = 1/2
P(B) = Probability of solving the problem by B = 1/3
Because A and B both are independent.
⇒ P (A ∩ B) = P(A) . P(B)
⇒ P (A ∩ B) =
P(A’) = 1 – P(A) = 1 – 1/2 = 1/2
P(B’) = 1 – P(B) =
(i) the problem is solved
The problem is solved, i.e. it is either solved by A or it is solved by B.
= P(A ∪ B)
As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
⇒ P (A ∪ B) =
(ii) exactly one of them solves the problem
i.e. either problem is solved by A but not by B or vice versa
i.e. P(A).P(B’) + P(A’).P(B)
=
=
⇒ P(A).P(B’) + P(A’).P(B) = 1/2
Question 15.
One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent?
(i) E : ‘the card drawn is a spade’
F : ‘the card drawn is an ace’
(ii) E : ‘the card drawn is black’
F : ‘the card drawn is a king’
(iii) E : ‘the card drawn is a king or queen’
F : ‘the card drawn is a queen or jack’.
Answer:
Given: A deck of 52 cards.
(i) In a deck of 52 cards, 13 cards are spade and 4 cards are ace and only one card is there which is spade and ace both.
Hence, P(E) = The card drawn is a spade =
P(F) = The card drawn is an ace =
P(E ∩ F) = The card drawn is a spade and ace both = ..(1)
And P(E) . P(F)
..(2)
From (1) and (2)
⇒ P (E ∩ F) = P(E) . P(F)
Hence, E and F are independent events.
(ii) In a deck of 52 cards, 26 cards are black and 4 cards are king and only two card are there which are black and king both.
Hence, P(E) = The card drawn is of black =
P(F) = The card drawn is a king =
P(E ∩ F) = The card drawn is a black and king both = ..(1)
And P(E) . P(F)
..(2)
From (1) and (2)
⇒ P (E ∩ F) = P(E) . P(F)
Hence, E and F are independent events.
(iii) In a deck of 52 cards, 4 cards are queen, 4 cards are king and 4 cards are jack.
Hence, P(E) = The card drawn is either king or queen =
P(F) = The card drawn is either queen or jack =
There are 4 cards which are either king or queen and either queen or jack.
P(E ∩ F) = The card drawn is either king or queen and either queen or jack = ..(1)
And P(E) . P(F)
..(2)
From (1) and (2)
⇒ P (E ∩ F) ≠ P(E) . P(F)
Hence, E and F are not independent events.
Question 16.
In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random.
(a) Find the probability that she reads neither Hindi nor English newspapers.
(b) If she reads Hindi newspaper, find the probability that she reads English newspaper.
(c) If she reads English newspaper, find the probability that she reads Hindi newspaper
Answer:
Given:
Let H and E denote the number of students who read Hindi and English newspaper respectively.
Hence, P(H) = Probability of students who read Hindi newspaper=
P(E) = Probability of students who read English newspaper =
P (H ∩ E) = Probability of students who read Hindi and English both newspaper =
(a) Find the probability that she reads neither Hindi nor English newspapers.
P(neither H nor E)
P(neither H nor E) = P(H’ ∩ E’)
As, { H’ ∩ E’ =(H ∪ E)’}
⇒ P(neither A nor B) = P ((H ∪ E)’)
= 1 - P (H ∪ E)
= 1- [P(H) + P(E) - P (H ∩ E)]
(b) If she reads Hindi newspaper, find the probability that she reads English newspaper.
P (E|H) = hindi newspaper reading has already occurred and the probability that she reads English newspaper is to find.
As we know
⇒
⇒ P (E|H) =
(c) If she reads English newspaper, find the probability that she reads Hindi newspaper.
P (H|E) = English newspaper reading has already occurred and the probability that she reads Hindi newspaper is to find.
As we know
⇒
⇒ P (H|E) =
Question 17.
The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
A. 0
B.
C.
D.
Answer:
Given: A pair of dice is rolled.
Hence the number of outcomes = 36
Let P(E) be the probability to get an even prime number on each die.
As we know the only even prime number is 2.
So, E = {2,2}
⇒
The correct answer is D.
Question 18.
Two events A and B will be independent, if
(A) A and B are mutually exclusive
(B) P(A′B′) = [1 – P(A)] [1 – P(B)]
(C) P(A) = P(B)
(D) P(A) + P(B) = 1
Answer:
Given: Two events A and B will be independent.
As A and B are independent events.
⇒ P (A ∩ B) = P(A) . P(B)
We solve it using options.
Let P(A) = a, P(B) = b
As, A and B are mutually exclusive
P (A ∩ B) = ϕ
Now, P(A).P(B) = a.b ≠ P(A ∩ B)
⇒ P (A ∩ B) ≠ P(A) . P(B)
Hence, it shows A and B are not Independent events.
(B) P(A′B′) = [1 – P(A)] [1 – P(B)]
⇒ P(A′∩ B′) = 1 – P(A) – P(B) + P(A)P(B)
⇒ 1 - P (A ∪ B) =1 – P(A) – P(B) + P(A)P(B)
= - [P(A) + P(B) - P (A ∩ B)] = – P(A) – P(B) + P(A)P(B)
= - P(A) - P(B) + P (A ∩ B) = – P(A) – P(B) + P(A)P(B)
⇒ P (A ∩ B) = P(A) . P(B)
Hence, it shows A and B are Independent events.
(C) P(A) = P(B)
As, P(A) = P(B)
Let we take the example of a coin
P(A) = probability of getting head = 1/2
P(B) = probability of getting tail = 1/2
A ∩ B = ϕ
P(A ∩ B) = probability of getting head and tail both = 0
Now, P(A).P(B) = 1/2 . 1/2 = 1/4 ≠ P(A ∩ B)
⇒ P (A ∩ B) ≠ P(A) . P(B)
Hence, it shows A and B are not Independent events.
(D) P(A) + P(B) = 1
Let we take the example of a coin
P(A) = probability of getting head = 1/2
P(B) = probability of getting tail = 1/2
Now, P(A) + P(B) = 1/2 + 1/2 = 1
But it doesnot inferred that A and B are independent.
Hence the correct option is B.
Exercise 13.3
Question 1.An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?
Answer:Given: An urn contains 5 red and 5 black balls.
Let in first attempt the ball drawn is of red colour.
⇒ P (probability of drawing a red ball) ![](data:image/png;base64,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)
Now the two balls of same colour (red) are added to the urn then the urn contains 7 red and 5 black balls.
⇒ P (probability of drawing a red ball) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAAgCAMAAACrZuH4AAAAAXNSR0IArs4c6QAAAFdQTFRFAAAA///b//+22////9uQ29u2kNv//7Zm27ZmkLbbZrb/25Bm25A6OpDbtmY6tmYAOma2kDqQOmZmkDo6AGa2kDoAAGaQZjoAZgA6ZgAAOgA6AAA6AAAAIEqLfwAAAAF0Uk5TAEDm2GYAAAB5SURBVHjazZNJDoAgEAQbdxF33Pv/7/SiBxMQD2qoc6WHmTTAZ4iBJPWd0YYQfeLISRvXIHeE9iHif2qebE8fxysfTfEHUWkAiCfS3LOimzSAYM6R2fZTZzmi1WWoEvdGZq37YSirIOoxPD5MKo0JJFkiWEhSvnG+HRgSCNykXTilAAAAAElFTkSuQmCC)
Now Let in first attempt the ball drawn is of black colour.
⇒ P (probability of drawing a black ball) ![](data:image/png;base64,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)
Now the two balls of same colour (black) are added to the urn then the urn contains 5 red and 7 black balls.
⇒ P (probability of drawing a red ball) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAAgCAMAAACrZuH4AAAAAXNSR0IArs4c6QAAAFpQTFRFAAAA///b//+22////9uQ29u2kNv//7Zm27ZmkLbbZrb/Zrbb25Bm25A6OpDbtmY6tmYAOma2OmZmkDo6AGa2kDoAAGaQZjoAZgA6ZgAAOgA6OgAAAAA6AAAAau0YfAAAAAF0Uk5TAEDm2GYAAACFSURBVHjazZNJDsMgEAQb78HGW/Aa9/+/mUuQksMwOdiS60qpRzANcB2O5FFEDatm2H+mDJqUvbQcM7Wx02ehZJiOPGrci56B+B6+4C8XTbkPpvMAkC9Si5px8QCS9YFKup/zoWe7ZjipZsGoxDJ/DCcKpp9TAOUAlFb6K2yRbCRpz3i+N9pNCeLeUW0bAAAAAElFTkSuQmCC)
Therefore, the probability of drawing the second ball as of red colour is:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYYAAAArCAMAAABLq99TAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAA///b//+22///tv///9u2/9uQ29u2ttv/kNv//7Zm27aQZrb/Zrbb25Bm25A6ZpDbOpDbtmY6kGZmtmYAkGY6ZmaQOma2OmaQkDo6AGa2kDoAZjpmZjo6ZjoAOjpmADqQADpmADo6ZgAAOgA6OgAAAABmAAA6AAAA/iFUoAAAAAF0Uk5TAEDm2GYAAAQaSURBVHja7VtrV9swDE1o07WhLMvmAUuAlbGO/P9fuJOUOH5LsmVOx6rP0r2S5VdkpSgucpGL/CPyefd/cIbl+hFwqRkmOazYkTfHEfh1lw6J95GPk9O+ehr+rMMqYj95vyeyI5CpoH5IvI98nEz25bdtsbnfNigWqbW5g5WxyPqQgMg25Ejk8DErJ01g+68t3vmrl9Yw40A2ZiYC2YDULBQfc3KSJWzfdASWZtlMyx5a1WhkY0hgZANyJrJ8zMjJnIbN7x2epeyVgGvAAI+8Of48Ds87NLIBKYlsH7NxcqdBdASWetir8YYXMh65/HFTVL260FpKSKLz+piNkzkNy0zCsAjtJhieQDTkcXq2+KmpQuqLwfAxEyd3GhrpNIJF9Ro8C0nIE1qHP2VVyEYbd8PHTJzMaVC4ESzCUAlNPAqyaM3xg6a0AmkMn0AOVhIndxrqZXOELxjWfKkDJhRk8borql71sgZMFEhdFXdbTeWMS4Pffs5Q+X38tv/1JXxAmzihDYCCXD0Ow3C/xW4tOqQ+yWrsWCVx0iVoX/b0ChHubpgPmZcoipNb1CMqRvzbaT5kZqLIOxarkK4ViDP7PZCZiWI4nZJQgEV+6uBOHa0ewIqcHAKiVJF4BicVYD0xXD2cEKuHNXMa4pDPLg1WGGkFWA9x2T+vp233jjIGWqysyMljh0sDfgW7wuBPw+n6gTj8yGmIQz67NLjCQKdBDLPMrntjKPvDp6duxZ8GOrLtdf40AJzOMNQ0wPbOGKTVsNAoD7qDQ3THLV42ZFt0TZvIQLJcS+e0wgBXg22Pm0oWC9dqiEE+v02JmoaUTemQbVM6fIxN6cC0KfnyP59AK8pURH03RCGf33eDI4yG/bvhrU4DX2jIaYhDPrs0uMJI+PjzVAKq21OWN7crwue/FisrcnIxA5cGfDHDCiOtACvTWV4/DeoDubdkoisiSnvsyIamOiOdSBk4IyRoL6vEYrgpqhf/bJAlE00RU+hmR7Y058FzI+XgjCkbBu3nc2Vqbmi82+xSMtEUMc8+/MgG5EzkQcrCGZGGoL32WBU87dSLwKyIfARlRjYg1ddWB1IWTu6CtvZ0GzyitGvxPNNxLQHMyAakp/cgfN4mcsYuCa+90l0S7nFWHJ8VsQ0y3MgGpLMTB2jYTuWMvJf690nZawW0rS2OS0Vsuxg3sgHp6kuDevBSOaMuSz109iAexKXjUhHbPMmPbEDaXZrg634yZ0wWOsSkFR2yWUoqYluJ+ZENSLtnGaJM54y5KwXtT13p0wZbt7DjUhHbWJ8B2YQ0O/hBSgZOqoD24x8W04QqRQsWXBZF5G8mGZBtyPl/FhMpJyfxeAbtxz+W3iqzPq2lZCIVsT9dZUC2IUciB1JWTupXdJr9RS7yAeUv2N7D3yao/L8AAAAASUVORK5CYII=)
Question 2.A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.
Answer:Given: let E1 be the event of choosing the bag I, E2 be the event of choosing the another bag say bag II and A be the event of drawing a red ball.
Then P (E1) = P (E2) = 1/2
Also P(A|E1) = P (drawing a red ball from bag I) ![](data:image/png;base64,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)
And P(A|E2) = P (drawing a red ball from bag II) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADkAAAAgCAMAAACfHyDkAAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OjoAOmZmOmaQOma2OpDbZgAAZgA6ZjoAZjpmZrbbZrb/kDoAkDo6kGaQkLbbkNv/tmYAtmY6tv//25A625Bm27Zm29uQ29u22////7Zm/9uQ/9u2//+2///bKbI9WAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA2UlEQVRIS92U2RKCMAxFU7e6FESsC1JRlvb/P9EWfXEmSXlgRsY8cjn03pIEYBr1zIRQjBV3W+Gq3V+gml1J9L7NCDIg7ZomAQxDmoS7OIasuJjcmYYHabeNB5sjbdcVyxpVbSp80aQJMnsP02iu0VwUIXBfaMdF5C8bn+9Q1iLyoEQjuh103j+81Pk9tMOn4R2vlfg8OJ1AlzLD4E4bnLSpf15wqybX1Jmq7gJNVKMcQULIqcic9lA6nZ8x3WY5OE267VeCWGBkK1nSZyDdVjLyVx5yXv6k/V7zyAuJkdmEPgAAAABJRU5ErkJggg==)
Now the probability of drawing a ball from bag I, being given that it is red, is P(E1|A).
By using bayes’ theorem, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEEAAABKCAMAAADT7sm0AAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAttv/tv//25A627Zm27aQ2////7Zm/9uQ/9u2//+2///bmT1U3gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABrklEQVRYR+1Wa0/DMAxMymMFCusGjG7L6Nq0zf//hyR9MD5gX1QziSHyYZqU9Ow7O+co9YtX9bwXZdeu9dW7BKHJDqUMwUf/RxhK8Ed0SEQ96bap1vr6VdKVF/Gtq9Za30i0Mnqj2lxy/cwitJ1++14wX4dxAUEthRBdCCNhEaI0KUFivFqwKV2x5LJtUoTgiiAmubr8BSGYRc0hmKUFCOWtB7AkD/tQA4TmzvejMxRCt9orgGCGjiEQepE9QskWixPBTi05G6FHR0rixpUjlKgfUBJeCykECnHaHwvr5WdiRh36xDz5S/hHpBJtQlFUohKMOvQ1nDveS6082am2kFSTt/IoccIh3ogx0XabHVgTA0S7XOuMt0F2ZvWxqxSPEzBxLGVhEzlE1E8kdqCwRIdhhMcqTdSFZmqLMBD4RRNtw0vsEQIgomz4WKI0SDRRGiKWKJLyfPs/a4LnyxMhB59MJE/7MtkodUR3k0ljfKUZfDEokAmBfe2hOxdYSNzeW5xw+Pf2RD79USH9vuvNZfidt7q811Dw9h+jC6rpJXiqRR3lKxmm/26eBhf51Qe/URzOWrhkigAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Question 3.Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?
Answer:Given: let E1 be the event that student is a hostler, E2 be the event that student is a day scholar and A be the event of getting A grade.
Then ![](data:image/png;base64,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)
and ![](data:image/png;base64,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)
Also P(A|E1) = P (students who attain A grade reside in hostel) = 30% = 0.3
And P(A|E2) = P (students who attain A grade is day scholar) = 20% = 0.2
Now the probability of students who reside in hostel, being given he attain A grade, is P(E1|A).
By using bayes’ theorem, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 4.In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4. What is the probability that the student knows the answer given that he answered it correctly?
Answer:Given: let E1 be the event that the student knows the answer, E2 be the event that the student guess the answer and A be the event that the answer is correct.
Then ![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
Also P(A|E1) = P (correct answer given that he knows) = 1
And P(A|E2) = P (correct answer given that he guesses) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAAgCAMAAAA7dZg3AAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOjoAOmZmOma2OpDbZjoAZjpmZrbbZrb/kDoAkDo6kGaQkNv/tmYAtmY6tv//25A625Bm29uQ2////7Zm/9uQ//+2///bL3fpFQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAcUlEQVQ4T2NgoBqQE2XDYZYEJw8uKQYGcfpJyYmwSmF3ojgjEHBTLSiIM0gEZCkYMAkT0gFViF8ZKQYSspDK8jLs/DhMlBPkwCUlzieAQ0qaSw6HlCyvmJwAnxC2eAZHMiMLjiSAy0CgsyXZmcXIDA8A0XIEOQegeXYAAAAASUVORK5CYII=)
Now the probability that he knows the answer, being given that answer is correct, is P(E1|A).
By using bayes’ theorem, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 5.A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?
Answer:Given: let E1 be the event that person has a disease, E2 be the event that person don not have a disease and A be the event that blood test is positive.
As E1 and E2 are the events which are complimentary to each other.
Then P (E1) + P (E2) = 1
⇒ P (E2) = 1 - P (E1)
Then ![](data:image/png;base64,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)
and P (E2) = 1 – 0.001 = 0.999
Also P(A|E1) = P (result is positive given that person has disease) = 99% = 0.99
And P(A|E2) = P (result is positive given that person has no disease) = 0.5% = 0.005
Now the probability that person has a disease, give that his test result is positive is P(E1|A).
By using bayes’ theorem, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALEAAAArCAMAAAAACpnSAAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZmZmZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtpA6ttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bLg5FNwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACWUlEQVRYR+2Za3eCMAxA283JdLqXbFO2MRXk///ENS1tCKbUg576OPDBkypLL1kJ9irEcFxDBapPKSc/jhSHXLR+kXKkT8Yo9kVW6cOmTO9/63lxyEX53VJUGZyMUWxgsb1bCbGVi3piHDLRbj5Tp8ErRtGBha7Ybj6uZ8YhExUJXBgUH6PoxDA9EMNrTWOG+AFGprIwxuhMxAbKEcPQvEUj9Q9Ziepbqk8wio2MYIcQQ/sYfcESwug8xG4d2xUw1tU1CxwjA1cksDZoFBO7vsEsBA65SIPl0F1aUUziHBobdjccchGAYV/BKCYw6QtqffrahLs318mTuUkFRlGBhSjVg3cKEKZkbshGRfK4tGvYRpF5h+mGCgwVGCpwmRXIpD3cs/MyQf1U7gouMri2amreG1gVV1n3AfpkFfizW4FWRip5TuR4wklxd8KIoyKB1uVpwFTy9Hc8VWqti2IJJ1UnFYlG4sSR8Rueg0qe/o6HEIeTwo7gDYhZcdRJTCVPf8dDiMNJVaOd6etixVEXMZU8nNk50PE0icNJ1e52stHErDgKE9uNJGd2GlKly/HsE3clFbvnlbGOrDhS+8NEjvhWQSVPgNjneNpP0kOSKuOhiPNFQxc10n+8izLlewU1OEc4Hm5VGAPKJd3aL151T+DEEb5HmwaVPP0dD3fn1ebIp4vMqtAHJ458RoZKnv6OhxCHkwImErfFUQbX6quxszr6r/o7HkIcTkoKuyeOMlX9xm8XrUeJtTpHOh5C7ASRPynYO/McZsRR+arW+RR/HzrZV5Uh0S1U4B8U0nYrOS2WsQAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 6.There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?
Answer:Given: let E1 be the event of choosing a two headed coin, E2 be the event of choosing a biased coin and E3 be the event of choosing an unbiased coin. Let A be the event that the coin shows head.
Then P (E1) = P (E2) = P (E3) = 1/3
As we a headed coin has head on both sides so it will shows head.
Also P(A|E1) = P (correct answer given that he knows) = 1
And P(A|E2) = P (coin shows head given that the coin is biased) ![](data:image/png;base64,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)
And P(A|E3) = P (coin shows head given that the coin is unbiased) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAAgCAMAAAA7dZg3AAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAb0lEQVQ4T72RSw6AIAxEi39FhYoi3P+ignHZCibE2b4y0w4AxeR1w3htneQQAP6HvKoPekUUQUOxKvKMVAy9Va2pF8/g+9gXw1RgUb5LIXrS0U0LGP78s2WbQfa7DB0V8pElNhA7Uyu6MXZPooyOLjHfArzqNRIZAAAAAElFTkSuQmCC)
Now the probability that the coin is two headed, being given that it shows head, is P(E1|A).
By using bayes’ theorem, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 7.An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively.
One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
Answer:Given: let E1 be the event that the driver is a scooter driver, E2 be the event that the driver is a car driver and E3 be the event that the driver is a truck driver. Let A be the event that the person meet with an accident.
Total number of drivers = 2000 + 4000 + 6000 = 12000
Then ![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
As we a headed coin has head on both sides so it will shows head.
Also P(A|E1) = P (accident of a scooter driver) ![](data:image/png;base64,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)
And P(A|E2) = P (accident of a car driver) ![](data:image/png;base64,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)
And P(A|E3) = P (accident of a truck driver) ![](data:image/png;base64,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)
Now the probability that the driver is a scooter driver, being given that he met with an accident, is P(E1|A).
By using bayes’ theorem, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHwAAAAqCAMAAABGF89mAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6Ojo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZpCQZpDbZrbbZrb/kDoAkDo6kGY6kNv/tmYAtmY6tpBmttv/tv/btv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///b5HsuLQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACSUlEQVRYR+1X3VrCMAxt8Y8pioKCytQqOhyzW9//6WzTlHZbmd3gGxeym/GVLCc5OUk2Qo5XXwyk87e+oKo4+YyefB0InI+XycHAZc5H8IMU/kj7P6R9cKgJJ14jSunpUzjn4uPaMxFFPCm5SMbLcJfBliK+W3uMq+CEXy2CfTYZiliSQ+lI5QIgeECp9b8BF/G5Do5f7qeaxXRISBqpHZDBIoAD8hN5wDNqJJRgFDsSIGKFlchE9S9zYx5w9hhh9bunzoFkvDRkRieE62QxBicnQ3vx8Gl4r8kgnILMaUCbOZ7CwTeQsIlP55sM7ZZkrkEAcjEFbcFl0QErjWQNN+DKwAMu4oVhhxC2KTrzOG2OJhmYkoK6B7dSxbiEIJrMA86v17YmFrwGZFO0v0pGLrjBcWkvGWOBE/CFr0YN4N60G2gH+zq4eEf9q5oX96q3M+x/W/O2tNcFp8DLapettrqZOeAZVFlPAXkrT9wAyaFJqdXQm+0xfaAjkfqGP2DyISrw3r3P3TCV3oxwYcKZ8Qr9YMFVzaSduimrgAm3ajeB65PDybxEbHPiXC1T6GaRzig9C4uittW2gP+x1VA8MlxGn0k+Dfx2qOxzwS6UyNrucwfc7I1wQRpLaOwOqrbguoWd7dw+iLZPVMBZIO1tYfz2PBpFVmaVUPYDsd2LeJEyi3F37bB8u8fJ9XtH/e2gu8vwJ3FysqHvlTTcTWtLpnLWmcMkzDp0TGtQ8wCT5c5jpXGYhAKC6evK53I+wIcFrts+wftKMhTnF2dJOHyt9YqhAAAAAElFTkSuQmCC)
Question 8.A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?
Answer:Given: let E1 be the event that item is produced by A, E2 be the event that item is produced by B and X be the event that produced product is found to be defective.
Then ![](data:image/png;base64,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)
and ![](data:image/png;base64,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)
Also P(X|E1) = P (item is defective given that it is produced by machine A) ![](data:image/png;base64,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)
And P(X|E2) = P (item is defective given that it is produced by machine B) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFoAAAAgCAMAAAClz5+XAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjqQOmZmOmaQOpDbZgAAZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkGY6kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bTBxh4wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABOklEQVRIS+1Viw6CMAzc8IkvFN9T5wvZ//+h7Q0kGAQlw8TEJqQjdLf2rqxC/M3s+g2RcByHTUELof/QedmaI8So3qWqRc7zLYfEofRWWMzwXmFakgWlQbdQtg4coQJxwko5Uj6a7DUAOVfkGw1xkhPLoM16Jcy6vMqPjsxnrbuV4hSiK+Ye5mVSWWjOlriOp5AytcIN7yduoQXp2dmyhpEvO6/pfiSXW7w4LYHGV9KQtDQ1mqSEECAzK1ci+2oLoUYsYvBjQngDYOlx1YD6ISk0BPSoXpvk6jEbn2puL2z5/B/Gs2UdrstZigboyJMvJw6Sfl+RLDIZimYjPS429XWgnvakQ1H3LygT3gEuIDADWHLuVevdQvMtSbrjtlQ/A90EIclQVF0rI7wbQtKhyLN0yXom3g36V1HulPQZzxpt788AAAAASUVORK5CYII=)
Now the probability that item is produced by B, being given that item is defective, is P(E2|A).
By using bayes’ theorem, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 9.Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.
Answer:Given: let E1 be the event that first group wins the competition, E2 be the event that that second group wins the competition and A be the event of introducing a new product.
Then P(E1) = 0.6
and P(E2) = 0.4
Also P(A|E1) = P (introducing a new product given that first group wins) = 0.7
And P(A|E2) = P (introducing a new product given that second group wins) = 0.3
Now the probability of that new product introduced was by the second group, being given that a new product was introduced, is P(E2|A).
By using bayes’ theorem, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIEAAAAqCAMAAACjpchPAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtpA6ttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bgDksKgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACQUlEQVRYR+1Y2XbCIBAFu5iqNdUuqdXUpmqU///BDosEDAzMOfXBczIPMcvNcJmFcGVssGAExIrz6bZ7tFtu5IXYLTh/UKdRy4aizkT1uD9Wdz9mmOOC6/Oav7Njae8HWBCgqLPDCOZ54B96iHa2bQyDMVw25/sBAgQoqzFntRzwVEqINsNAnVtm4UQQoHFnMgmSgTz2GSh6cXMZJKAyr2FnmoE+9hi0hUlOhITDIAWFBEecYQxENcci4CYsCYVJRpydsxCoA1F1NxN1kIYiCFOJHUEb2npsM5PKQhqKIFTDuTV/ZtDI0jigeciHYs5kG6hUnDuykQsEFM4T/IgaZ5ALxZ3B0sZnqiMh7WJVcM7v3+SaqCzOgABNO8Mrfng6RGCIwDUiYLocGl0vJtlGeJEAlcPrdQezCMnQK7H5dNjsGV8RSIxQx4TwIgF6xZnevuvf/D4hQKNx8TRTKz/OtlPNViH8KgHKmBJgjjBzXfqaydvQtgW2aBCgrBmtYbcT2fpfaCZnS30qX3MZJKCnUm509LFvvmZyJ1bPFbuYkaGuKHGcXmgmx+1hus9lkILq2aMM7MO2mBRas59eNizBIBsKgm3DxDfvhFkvBpaB+ATNXkHwlcQBBk1UtxGgsAGGVvhy5HEgC648aos5KAhjuHIkQEE4opXoPrRaHs2CmgYBCi0ZrmtfMymFYsmiDAhQn+xlc/maqQae3X866JpIgMoxd8VzTIZ6mum4hPTPzPIJtYAsCASoDOtkffvfy/+ZwR+G1z435rIPiQAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Question 10.Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?
Answer:Given: : let E1 be the event that the outcome on the die is 5 or 6, E2 be the event that the outcome on the die is 1, 2, 3 or 4 and A be the event getting exactly head.
Then ![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
As in throwing a coin three times we get 8 possibilities.
{HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}
⇒ P(A|E1) = P (obtaining exactly one head by tossing the coin three times if she get 5 or 6) ![](data:image/png;base64,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)
And P(A|E2) = P (obtaining exactly one head by tossing the coin three times if she get 1,2 ,3 or 4) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAAgCAMAAAA7dZg3AAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAb0lEQVQ4T72RSw6AIAxEi39FhYoi3P+ignHZCibE2b4y0w4AxeR1w3htneQQAP6HvKoPekUUQUOxKvKMVAy9Va2pF8/g+9gXw1RgUb5LIXrS0U0LGP78s2WbQfa7DB0V8pElNhA7Uyu6MXZPooyOLjHfArzqNRIZAAAAAElFTkSuQmCC)
Now the probability that the girl threw 1, 2, 3 or 4 with a die, being given that she obtained exactly one head, is P(E2|A).
By using bayes’ theorem, we have:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUYAAAAvCAMAAACohZPkAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6Ojo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZpCQZrb/kDoAkLbbkNv/tmYAtmY6tpBmttv/tv/btv//25A625Bm27Zm27aQ29u22////7Zm/7aQ/9uQ//+2///b5ZKXzwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFAUlEQVRoQ+1b63rTMAx1NtiAwUbHWKFcFgYLbd7/AYlsy5YiO4kTZ3z1mh/rvnRS7RNZl9MzpU7XCYETAgkItN+vHgJ/Xn/4EXPSt2i3H0N/OuAhYXlH8qft9uYpuNT92/vwFoRFBEYV9XAk0Awvs91WcL3T0WYgsLeqiiK3f/MVHVETD1q7faWfgL4TckE8FIEc38Rhc6HU78tzOMuN/qn0LfX3kgVgbVBy7xsTawG21ZkG2gAbckE8lIdjuwXMagg986t73TEYWTiiCVp0RrvPl4Cfi2jtirkoOhwNEk3VYbC38UfA8WFDcp43QYsu/D79NKcaE4N5IvSKZc0iQpNEY2OOpYnKxz4OO3fDm6BFF84XqtYpgcDYd+E9FIFcL0Z0boRI8jBC1REwuuSoYdQmDsZ2e2+DmZapnoudT6/F4aiL6tk1nEcTTjYamyEY0QQt1P7qyaZWEo19F2XD6PBih1rEiwfB504XjbVum+AxxHOjerEw/rmrqmsLaD83wm2E8XALWbUx5R4KtkeauCg+NxqkepV6d3+4/aIebd05bAAef+iZRaOznm4WGYzMBfFQXGq0jTKDx/TOFlTs9uDVvEFMbNDZl8OmO9W0/eYuSu4bocK4AmpmEpzkTBjWNghhBjEwUhNtcdhoH/By/sCGQeZilSkmwqX0e9RnZUYCDfKjTY2RUJIW8o51sUowxrgUsYpnZUbEqmpEcT7DY11k24dkRgJEiIMRGRO1ylOM5fjeGWk6FJuu/C7gG62LjKeKkClRLsXBiIxJt4f/1vxDtuN8WXKJzeCi/5l+Fo1zKQ5GZEy6kumpvuRdlGggmRHJpSCMjjHBTqxEQObtSTIjkktBGB1j0n1Uyd3/DCQlM2JqDCUBLIyeMQEYXXLc6dEVLjtbzFjF0Zt4MiXOpVgYPWPCYBQQOFhfzC90Xo9zKRZGz5gMw3j0oTVjA5JgkkSIHUk9Y8Jy4+lQ62EUs2CcSzEwEsYERlY71s54ciWaWIKEACq5FDvZWyZFUwXT+8a1NA8pI8haa3CTlGBGQlwKwMgYk4QpZj3Nw/SBeL01hHQXUS5FvjEYjJPmdBbQ45qHRMlDLt2FIxEAgjHlBiarGJeSyvBMmdOBNUzRPIxLHqCtdVcm3YUjETQEY8oNhyPXbiGXkso3TpnTeX6VFtAMUM2DLYUDkgcGYybdhSMRaHmIKjfCtXM2ETJlTtc1zVX7cc1DSDXBH28AxqW6C08iUBg5WoFUmKkVmTKnw0elaB6Cqgk22MeicYHuwpMIHsYB5UYm+HyO7SsY5JyuYUzQPAQ9sC+ZJYxLdReERCDf5sQlB9lh7CsYdCwt0jwEPTgYxSSVRXdBSAQfjQO7yA4jPrF8moegomwkGs225q+BkAjx3Lie7mJoTlftN3cqUjQPBEbvYSw3RmCcqLsgsgvU6+LSwx5Wi0Yxp6tf7+9wLUmaB9/weA98sA+UGL2v2boLSiLwhiei3MgM4/Cc3pU/cyVpHuikjx74LCXbb/MxGMepugsquzDRGPWQGT9ctxc9BDQPDsYUzQNTTaAH/vUkhTGD7oKTCHQYlMqNVWCkTgPNaTiWnJG0EHesh6ksU5QriBJVE9YwotzIDaxkVxCEBZoHnRYWMDzLdRejHrLj2Psfq3b3Wqto5/+PlfGwgG9crruY4CE3jtyf7sW6dD3/WuxhNlfg1rzcw/ztF2z5D2YZzPhUvi3dAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 11.A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
Answer:Given: let E1 be the event of time consumed by machine A, E2 be the event of time consumed by machine B and E3 be the event of time consumed by machine C. Let X be the event of producing defective items.
Then ![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
As we a headed coin has head on both sides so it will shows head.
Also P(X|E1) = P (defective item produced by A) ![](data:image/png;base64,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)
And P(X|E2) = P (defective item produced by B) ![](data:image/png;base64,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)
And P(X|E3) = P (defective item produced by C) ![](data:image/png;base64,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)
Now the probability that item produced by machine A, being given that defective item is produced, is P(E1|A).
By using bayes’ theorem, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 12.A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
Answer:Given: let E1 be the event that the drawn card is a diamond, E2 be the event that the drawn card is not a diamond and A be the event that the card is lost.
As we know, out of 52 cards, 13 cards are diamond and 39 cards are not diamond.
Then ![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
Now, when a diamond card is lost then there are 12 diamond cards out of total 51 cards.
Two diamond cards can be drawn out of 12 diamond cards in 12C2 ways.
Similarly, Two diamond cards can be drawn out of total 51 cards in 51C2 ways.
Then probability of getting two cards, when one diamond card is lost, is P(A|E1).
Also P(A|E1) =12C2 / 51C2
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, when not a diamond card is lost then there are 13 diamond cards out of total 51 cards.
Two diamond cards can be drawn out of 13 diamond cards in 13C2 ways.
Similarly, Two diamond cards can be drawn out of total 51 cards in 51C2 ways.
Then probability of getting two cards, when card is lost which is not diamond, is P(A|E2).
Also P(A|E2) =13C2 / 51C2
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIUAAAAqCAMAAACqTmg1AAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmZmZmaQZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2//+2///bwtdwSwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC6klEQVRYR+1Yf3PUIBANrbWovevFqz9qPFussacXEr7/pxN2gYUkQNrpzeh4/JPM5O3y2F3IPqrqNAoR2N/cG4R6fMvYBbwmhgPutwUg2KsQpr7z14eM637Lzn+Y7+LsruqbsyQNArLbqq/RKDNEAOv4+ncOK1cPrWVxqXEt+4Lo4QPO0ttnlQImnQvy1/HbAmU9c7As4d5Vc2FoDPWa7ENg5+jm3QNsqA2hwiDn/W714MCq0YmM7WfpFtJiVrWIsXM+1IytqIQ0jV/by7CkAhaS28zlSSBMnH/T9bzOVWeUkT0PsqMaFtc1sVDNphRh8x1h2s/1odrzvEmcbsKmWahmQaLN9ACD3OqIZLdqVJ2SGENGIktPV0SJyuwSSIJlES52xgQ/Y/SokFx1Btl0flrDrSvmxMNw45ViASeVMgdWj8GjBUSbpMUjTb7RDyVKLAgmuY7dY25rqx1njL36rM8n8+YLuf+EdKR9VgQU2kCPEosAJs2Rf5fM3OnDKQKnCPyNEbB7WO/2dE/1VN4v4BMPoGeOFN+8u6eu8lj4F4jehNoxfB5r/f+O35/wy14wFgMTvhL20vzY3b614mvWQwBUO8aufLOemA+bERJo0URjm6CZ9uIrwcJ13aYL65uCNpMcWJBAy3bt9JHEl2Ex1Wa+9++M/4LGGOqPyMILtIUstE3QoE60GXmBNrKgt8QGuOIAxs9igZ1zMJf3gi31UOca++7qELAA2gUW77i/L4ia9ZE2k9wCkYVt7+dLc3h/XxELnN/bz5mor/oawN0XxJIhVkUeWGYBokKzaKGerY6LJkrUvzaL6wLMRwrRLGhjo5Cri879zgyLSMcFomtKxLmcZmSUfADa6kTiyeEyEum4FHcQN45iyGKszQgINy3F2wDLggRaONGYuwBFZo8gK74wmxAGIk9Asz2yxYnZhZ1KAi2aaMyiv9FJhLsTEl8GM9FmBKz0IRtedMwmRdeGoUECLbDPp/J/+voH68ZV95w/Aq8AAAAASUVORK5CYII=)
Now the probability that the lost card is diamond, being given that the card is lost, is P(E1|A).
By using Bayes’ theorem, we have:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUYAAAAvCAMAAACohZPkAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6Ojo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZpCQZrb/kDoAkLbbkNv/tmYAtmY6tpBmttv/tv/btv//25A625Bm27Zm27aQ29u22////7Zm/7aQ/9uQ//+2///b5ZKXzwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAE2ElEQVRoQ+2b62LTMAyFnQ1WYDDaMVYol4WNhSbv/4D4IsWyZcd20vWStT/YpZGqfFEc63AmxPl1JnAmUECg+3n9wA+vP/2K5vAiuvUydOhQhoLyTuTQbv35OVTq9v195Az8iAhGEc9wImgSZXbrSr0+qH4zDOA3VUXQbd997/PwCB30Rl8AnSKVYR7k3LNoV1dCPC0u5c3cqH+E0L8R/xa0A2tDSb94hAytLjRocyVSGWbIsVsraLXsPfOdQqG/bihGtx39CHnw14Xi1zf0cIbZYmyqpdhC/yFOerJ0zTPv0wjRfvlt7mpcF8wFiWWYLUbVjY25L003PnogNvZn278YIeorUesVgWAcyDBXjE8L2UsWo3ro+Bjt4qgxehH30Mv0KRXPMEuMktnFjbwjTT9BNzZDGP2I7fUzrKikGwcyzBIjAnNuav9MN1436vcxotabJnUV4mujIBleG8buBzJmayPB2N6qRbVRG02GMZhh1hi9J7Xc8Pz5eAcY25Xez9ibXn8HEY3uVL1ZdDBGM8wQo9kq92si8gBE8hFsgKkxBg71I2CD1K7kXU233wayk2GG/JBdVeGyZ6YYHOX09gch6CnG8FPv04h2pX9WXy4fnGGQZ9g5xrCWAlfTftpepZGArgAY6RBDUbAIniKRYRrYmJbCytirNMKrMhCmKDypDKUcs7QUxIiKCaxKpZ819nj/Huk2b9WGcoLemM5QWmuOloIYUTFRyxMRV0o/cuLxej+4nJJkegb26TlaCmJExQQfklNOZWaxXBnhWgpg7BUT/WSc1A8zg4g63rCWAsx6xURBIAPE7JiMOKGQMuJrKTgFoGKiMeLiuNGjq3qBpDWiiNMP0Y/qhJZiMFrFxMHIEfRcX803vTgvYcS1FIPRKiYJjKffW+VnYB8oAYyghGiMRDFx1sbzTa1wWIxxLUVjJIqJDKPiSvm1m19EjpaiMFLFpGjfeASeh4hUMDAHscEp4bvwlZGgliIxOoqJWihzp5gj8DzESohP5SyixHcR1VL4GzFxRd+tWXO6EQvNiwbYjf0Uz0NeCbEa7PmOqiGmpZQqPDlzutPQJABdEuWeh25NrAB5JdCbKmS7GOm7CGspxXpjzpzuqEQ2gDzySj0PDsa8Etx25LaL3fguxiohOXO6M5bbgN4lUe55CGBM2S5ogwRsF+U17HQLwh0MCc8D6cbxnodINw7aLog0ELBdHNh3kTOny+tm/4/YBpBZqtTzwDGmbRfBGvorqTKa3fRBfBc5c7qHEQf73iVR5nlgo1RmCQ5GX1wgKsJBfBeDA6ZdPrxuNG+M9zwEbmqaMGRJY3eEUwJREQ7iu8iZ01U39o4ccoqAcYTnIRvj37uquoGrGaoBr+RADSTFywmvOXO6M5YTjOM9DxGMTCpob7+JR9BGiTTAi477LkiKFxQXcuZ0x/NgA3oPmO7UIs8D337DiGSa3nwGQMURivguWNHDvgvccg7Oc5P2P1lzOkwx4LGxlgc9xYzyPFCMiRJEvTRnaH0XLCJRA6bIFhcmIoWCbZZ9ex4CUsEjLI2RVuIR/DeQ4uWa0cV+BJ4HVkKNFGN/WZNWeCDF/uwjR+B58EpoJMVGbqon6I2QYq9mJqc9x87pZFGY6JpQi57zZ0rlq9YOUpR/6Pwj/gNlfssDylqqVAAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 13.Probability that A speaks truth is 4/5. A coin is tossed. A reports that a head appears. The probability that actually there was head is
A. 4/5
B. 1/2
C. 1/5
D. 2/5
Answer:Given: let E1 be the event that A speaks truth, E2 be the event that A lies and X be the event that it appears head.
Therefore, ![](data:image/png;base64,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)
As E1 and E2 are the events which are complimentary to each other.
Then P (E1) + P (E2) = 1
⇒ P (E2) = 1 - P (E1)
⇒ P (E2) ![](data:image/png;base64,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)
If a coin is tossed it may show head or tail.
Hence the probability of getting head is 1/2 whether A speaks a truth or A lies.
P(X|E1) = P(X|E2) = 1/2
Now the probability that actually there was head, give that A speaks a truth is P(E1|X).
By using bayes’ theorem, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHEAAAAqCAMAAACz6UTWAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OgBmOjo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmY6ZmZmZrbbZrb/kDoAkDo6kNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///b7ZVa1wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACKUlEQVRYR+2W2VbCMBCGU1yoCgrUlWpUwlJD8/6v58wkaVMKNF1OvDEXwEnT+Wb5ZwJj/2vQDIjRx6D2Go3JODAxT14DE/kiC0vMpruwxPzxgwUlqnTBkCiWjQIb6EAWmRWMSI4HzeofEYPPHKilZ0Oqr+mqLiBSX7nEw/dAImNMpfPdEWMHRCbv+shQpaTlCbpNls1GFJVWaV/AqSWDdF3tmLzpczPkyZixbXwB2czwg9EG+4kPiGwTYW75GLMgANt5qRQBAvzXv+wXPySqFBySt1TntkFKyqFZmpNBAFKHZcBOCKaOMh6rVPtxWNnGeN2JUMZodmljQ+EWTmmtitHLzGxx9/lJXp7Yied2CwG2MRSmIOKxY8Q8oULj4kUheWH0fAeKkS0TaXM0AyUIbY9cyI4R4ahtzJJYC7AMq/LIJVrjblYrh23VxJzbIM8QnVfPZJVO1Ynq06iXQpOTFYjHZLXIgV9W68ohixWtQnes758cIukUuwhWnlTm3knp2AeV7tANXxTQTgCNFw6Ruh7Eg53Vth9dl1A4Vnc0c+yUI+WVRNyGuCCH+MBj5qx9BmG9r90YHT/PhyhjlKvfbVW7O44TG+4Oo4fG4upSVu9Hxa9xYre8H1sRq37hBVX0vJ/Ljub9X+h9skeMHdkynsTRpY9UOwJqr6n3N7ZP/bQ6FJMGWLup1B9djrL+thotcIwuaIwcSrjHP0XB1v4Z2njAf9HBHO8C+gWYoTFGDESgbgAAAABJRU5ErkJggg==)
Therefore correct answer is (A).
Question 14.If A and B are two events such that A ⊂ B and P(B) ≠ 0, then which of the following is correct?
A. P(A| B) = ![](data:image/png;base64,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)
B. P(A|B) < P(A)
C. P(A|B) ≥ P(A)
D. None of these
Answer:Given: A and B are two events such that A ⊂ B and P(B) ≠ 0
As A ⊂ B ⇒ A ∩ B = A
⇒ P(A ∩ B) = P(A)
As A ⊂ B ⇒ P(A) < P(B)
As we know ![](data:image/png;base64,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)
consider
![](data:image/png;base64,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)
it is also known that P(B) ≤ 1
![](data:image/png;base64,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)
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⇒ P(A|B) ≥ P(A) …(3)
Hence, the correct answer is (C).
An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?
Answer:
Given: An urn contains 5 red and 5 black balls.
Let in first attempt the ball drawn is of red colour.
⇒ P (probability of drawing a red ball)
Now the two balls of same colour (red) are added to the urn then the urn contains 7 red and 5 black balls.
⇒ P (probability of drawing a red ball)
Now Let in first attempt the ball drawn is of black colour.
⇒ P (probability of drawing a black ball)
Now the two balls of same colour (black) are added to the urn then the urn contains 5 red and 7 black balls.
⇒ P (probability of drawing a red ball)
Therefore, the probability of drawing the second ball as of red colour is:
Question 2.
A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.
Answer:
Given: let E1 be the event of choosing the bag I, E2 be the event of choosing the another bag say bag II and A be the event of drawing a red ball.
Then P (E1) = P (E2) = 1/2
Also P(A|E1) = P (drawing a red ball from bag I)
And P(A|E2) = P (drawing a red ball from bag II)
Now the probability of drawing a ball from bag I, being given that it is red, is P(E1|A).
By using bayes’ theorem, we have:
Question 3.
Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?
Answer:
Given: let E1 be the event that student is a hostler, E2 be the event that student is a day scholar and A be the event of getting A grade.
Then
and
Also P(A|E1) = P (students who attain A grade reside in hostel) = 30% = 0.3
And P(A|E2) = P (students who attain A grade is day scholar) = 20% = 0.2
Now the probability of students who reside in hostel, being given he attain A grade, is P(E1|A).
By using bayes’ theorem, we have:
Question 4.
In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4. What is the probability that the student knows the answer given that he answered it correctly?
Answer:
Given: let E1 be the event that the student knows the answer, E2 be the event that the student guess the answer and A be the event that the answer is correct.
Then
And
Also P(A|E1) = P (correct answer given that he knows) = 1
And P(A|E2) = P (correct answer given that he guesses)
Now the probability that he knows the answer, being given that answer is correct, is P(E1|A).
By using bayes’ theorem, we have:
Question 5.
A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?
Answer:
Given: let E1 be the event that person has a disease, E2 be the event that person don not have a disease and A be the event that blood test is positive.
As E1 and E2 are the events which are complimentary to each other.
Then P (E1) + P (E2) = 1
⇒ P (E2) = 1 - P (E1)
Then
and P (E2) = 1 – 0.001 = 0.999
Also P(A|E1) = P (result is positive given that person has disease) = 99% = 0.99
And P(A|E2) = P (result is positive given that person has no disease) = 0.5% = 0.005
Now the probability that person has a disease, give that his test result is positive is P(E1|A).
By using bayes’ theorem, we have:
Question 6.
There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?
Answer:
Given: let E1 be the event of choosing a two headed coin, E2 be the event of choosing a biased coin and E3 be the event of choosing an unbiased coin. Let A be the event that the coin shows head.
Then P (E1) = P (E2) = P (E3) = 1/3
As we a headed coin has head on both sides so it will shows head.
Also P(A|E1) = P (correct answer given that he knows) = 1
And P(A|E2) = P (coin shows head given that the coin is biased)
And P(A|E3) = P (coin shows head given that the coin is unbiased)
Now the probability that the coin is two headed, being given that it shows head, is P(E1|A).
By using bayes’ theorem, we have:
Question 7.
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively.
One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
Answer:
Given: let E1 be the event that the driver is a scooter driver, E2 be the event that the driver is a car driver and E3 be the event that the driver is a truck driver. Let A be the event that the person meet with an accident.
Total number of drivers = 2000 + 4000 + 6000 = 12000
Then
And
And
As we a headed coin has head on both sides so it will shows head.
Also P(A|E1) = P (accident of a scooter driver)
And P(A|E2) = P (accident of a car driver)
And P(A|E3) = P (accident of a truck driver)
Now the probability that the driver is a scooter driver, being given that he met with an accident, is P(E1|A).
By using bayes’ theorem, we have:
Question 8.
A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?
Answer:
Given: let E1 be the event that item is produced by A, E2 be the event that item is produced by B and X be the event that produced product is found to be defective.
Then
and
Also P(X|E1) = P (item is defective given that it is produced by machine A)
And P(X|E2) = P (item is defective given that it is produced by machine B)
Now the probability that item is produced by B, being given that item is defective, is P(E2|A).
By using bayes’ theorem, we have:
Question 9.
Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.
Answer:
Given: let E1 be the event that first group wins the competition, E2 be the event that that second group wins the competition and A be the event of introducing a new product.
Then P(E1) = 0.6
and P(E2) = 0.4
Also P(A|E1) = P (introducing a new product given that first group wins) = 0.7
And P(A|E2) = P (introducing a new product given that second group wins) = 0.3
Now the probability of that new product introduced was by the second group, being given that a new product was introduced, is P(E2|A).
By using bayes’ theorem, we have:
Question 10.
Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?
Answer:
Given: : let E1 be the event that the outcome on the die is 5 or 6, E2 be the event that the outcome on the die is 1, 2, 3 or 4 and A be the event getting exactly head.
Then
And
As in throwing a coin three times we get 8 possibilities.
{HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}
⇒ P(A|E1) = P (obtaining exactly one head by tossing the coin three times if she get 5 or 6)
And P(A|E2) = P (obtaining exactly one head by tossing the coin three times if she get 1,2 ,3 or 4)
Now the probability that the girl threw 1, 2, 3 or 4 with a die, being given that she obtained exactly one head, is P(E2|A).
By using bayes’ theorem, we have:
Question 11.
A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
Answer:
Given: let E1 be the event of time consumed by machine A, E2 be the event of time consumed by machine B and E3 be the event of time consumed by machine C. Let X be the event of producing defective items.
Then
And
And
As we a headed coin has head on both sides so it will shows head.
Also P(X|E1) = P (defective item produced by A)
And P(X|E2) = P (defective item produced by B)
And P(X|E3) = P (defective item produced by C)
Now the probability that item produced by machine A, being given that defective item is produced, is P(E1|A).
By using bayes’ theorem, we have:
Question 12.
A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
Answer:
Given: let E1 be the event that the drawn card is a diamond, E2 be the event that the drawn card is not a diamond and A be the event that the card is lost.
As we know, out of 52 cards, 13 cards are diamond and 39 cards are not diamond.
Then
And
Now, when a diamond card is lost then there are 12 diamond cards out of total 51 cards.
Two diamond cards can be drawn out of 12 diamond cards in 12C2 ways.
Similarly, Two diamond cards can be drawn out of total 51 cards in 51C2 ways.
Then probability of getting two cards, when one diamond card is lost, is P(A|E1).
Also P(A|E1) =12C2 / 51C2
Now, when not a diamond card is lost then there are 13 diamond cards out of total 51 cards.
Two diamond cards can be drawn out of 13 diamond cards in 13C2 ways.
Similarly, Two diamond cards can be drawn out of total 51 cards in 51C2 ways.
Then probability of getting two cards, when card is lost which is not diamond, is P(A|E2).
Also P(A|E2) =13C2 / 51C2
Now the probability that the lost card is diamond, being given that the card is lost, is P(E1|A).
By using Bayes’ theorem, we have:
Question 13.
Probability that A speaks truth is 4/5. A coin is tossed. A reports that a head appears. The probability that actually there was head is
A. 4/5
B. 1/2
C. 1/5
D. 2/5
Answer:
Given: let E1 be the event that A speaks truth, E2 be the event that A lies and X be the event that it appears head.
Therefore,
As E1 and E2 are the events which are complimentary to each other.
Then P (E1) + P (E2) = 1
⇒ P (E2) = 1 - P (E1)
⇒ P (E2)
If a coin is tossed it may show head or tail.
Hence the probability of getting head is 1/2 whether A speaks a truth or A lies.
P(X|E1) = P(X|E2) = 1/2
Now the probability that actually there was head, give that A speaks a truth is P(E1|X).
By using bayes’ theorem, we have:
Therefore correct answer is (A).
Question 14.
If A and B are two events such that A ⊂ B and P(B) ≠ 0, then which of the following is correct?
A. P(A| B) =
B. P(A|B) < P(A)
C. P(A|B) ≥ P(A)
D. None of these
Answer:
Given: A and B are two events such that A ⊂ B and P(B) ≠ 0
As A ⊂ B ⇒ A ∩ B = A
⇒ P(A ∩ B) = P(A)
As A ⊂ B ⇒ P(A) < P(B)
As we know
consider
it is also known that P(B) ≤ 1
⇒ P(A|B) ≥ P(A) …(3)
Hence, the correct answer is (C).
Exercise 13.4
Question 1.State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
![](data:image/png;base64,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)
Answer:Given: A given table with values for X and P(X).
As we know the sum of all the probabilities in a probability distribution of a random variable must be one.
![](data:image/png;base64,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)
Hence the sum of probabilities of given table = 0.4 + 0.4 + 0.2
= 1
Hence, the given table is the probability distributions of a random variable.
Question 2.State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQ0AAADFCAYAAABO3qdlAAAACXBIWXMAAAsTAAALEwEAmpwYAAAScUlEQVR4Xu2bCawsRRmFfYqAiIgI4oLCA0TZNUpUQBGUKBoji4oRJAgoKhBJUFTiLoYAAsYFIawSTEhAUBBQEuWxRCCCbAJhvxjEsMoiq+D1HOgmbfvXTPWd6Z6ZzlfJedP9V3XXX1/3nK6uue8FL6BAAAIQgAAEIAABCEAAAhCAAAQgAAEIQAACEIAABCAAAQhAAAIQgAAEIAABCHRBYFFGJ/MZbWgCAQhA4HkCfTGNWRgHOY7nizcLHKORzkLe8y+MMicGAQhAIEUA00iRIQ4BCIQEMI0QC0EIQCBFANNIkSEOAQiEBDCNEAtBCEAgRQDTSJEhDgEIhAQwjRALQQhAIEUA00iRIQ4BCIQEMI0QC0EIQCBFANNIkSEOAQiEBDCNEAtBCEAgRQDTSJEhDgEIhAQwjRALQQhAIEUA00iRIQ4BCIQEMI0QC0EIQCBFANNIkSEOAQiEBDCNEAtBCEAgRQDTSJEhDgEIhAQwjRALQQhAIEUA00iRIQ4BCIQEMI0QC0EIQCBFANNIkSEOAQiEBDCNEAtBCEAgRQDTSJEhDgEIhAQwjRALQQhAIEUA00iRIQ4BCIQEMI0QC0EIQCBFANNIkSEOAQiEBDCNEAtBCEAgRQDTSJEhDgEIhAQwjRALQQhAIEUA00iRIQ4BCIQEMI0QC0EIQCBFANNIkSEOAQiEBDCNEAtBCEAgRQDTSJEhDgEIhAQwjRDLTARXV5YXSXfORLYk2RaBU3TieWmDtjpYyHmd0KCyRJWPSG73oHRm0fhF+vQN7TrHzy7ik/oYNo4yrxW1cYx0o3RDkfcbO0o6N8dPKp856Vapa9PIydE38M8Lftfq83rpSOnVUhclJ8e1lMhh0l8K3aTPP0pbdpFgoo+cvKuHbqodH2N1ZRpZOeY02kRJPy1dJtksyrK8Nq6R1qjEJrWZM45FSu5C6RxpGcn7B0t/l1bpIPGcHJ3XEukN0mnSNJrGVcrLDFcomJndFdLtku+JtksOx72VxF3S2kUyvm+PkJ6QNmw7wcT5c/IuD/W9eal0ljSTpuGBHFokv18FyNHa3rOyP8nNnAuyXXABllXsIcnm0XbJydE3S/laOc2m8ZYarB207/Ht1DbEop9h3fhaf6HWaLXi2P2HHdxSfc71L7v+lDY8M9qjyHnmZhoeyEukm6XHJE/9tpbOc8WUlJwLcrJyvSfI91zFbgni4w7l5Fjtc1pNY+kAzLsU8/j8hG+7NOVY5rNekeNubSeYOH9u3n6Q3S69VercNMa5EPp4MQAP6ATp8GI/wWcqwxspK1+MenFsTeml9Qr2QwJPBdF1itgFQd00hBYriR9Lfj395TQkNCAHz+bN8coBbVqrGqdpOEkP5Cjp3dL50t9ay7ydE6+s0z4cnNoxvxasFNQRGk7A7D4r+cvohdFpKl7k9rrQbZKv8yekJ6cpwVouXkzeRzpgUjmO2zQ8Dj+Vn5F2lV4/qYHR71QR2FfZ+FepvaYqq+eS8Su11zJeKd0t2dTePoV5lin9QBt+MHsRd2pL7nuWB/AmySvnH5V8nFfQp6XkjMO/9HhFul78c+F/pLZfT3JyrOY2rWsa1Rz95C6/mHWube035Vjm4V9QvHb1p7YSG3LeYXlvrOM9e1+ucp7O1zSGjOHZ6mEDKc/hWcvFkn87djle8rG7FPuT/sgZBwuhw69SDsfyLP7FxH/v4p+Huyw5OXrh3q9N9XKGAv7ZdRJlWN5fVlJeqJ+r6D5t+zj/WYDjO0ptlmE5Ptt3ViO1889U/mOZsng66inU/dKqlfikNnPGsb2Sc7v1K0n67yL8x2nT8pNrld80zzTMsm4Y/gXl29UBtLSdc639gNs86P9yxSY19c/Ju57yzM401tVIrpbs3tWyrXYM4tRafBK7ORfETx6vnvuvV8ufDQ/Stm+iafnjriq7aTUNG4Z/TfNi3c4VHaLtE6sDaGk751rbNHytX1XJwQuMPnYW/k6jTHsmTeMSZe8nseUnS1m20sac5LUAX4g7pG2er+1+I+dGclaeIR0r+c+K/WfkXpcpfy5sO+vcHI9TInPSo5L/Etfb/nPoLkpOjn7vdrtIJ3aQZE6OnmU4l+skP/Cul2wiH+8gv1QXOXmXx26mjTmp/npSXe9I9TNKPCvHrEajZNHRsbMwDnIcz80wCxyjkc5C3vNt/OQawSAGAQj0hACm0ZMLyTAg0BUBTKMr0vQDgZ4QwDR6ciEZBgS6IoBpdEWafiDQEwKYRk8uJMOAQFcEMI2uSNMPBHpCANPoyYVkGBDoigCm0RVp+oFATwhgGj25kAwDAl0RwDS6Ik0/EOgJAUyjJxeSYUCgKwKYRlek6QcCPSGAafTkQjIMCHRFANPoijT9QKAnBDCNnlxIhgGBrghgGl2Rph8I9IQAptGTC8kwINAVAUyjK9L0A4GeEMA0enIhGQYEuiKAaXRFmn4g0BMCmEZPLiTDgEBXBDCNrkjTDwR6QgDT6MmFZBgQ6IoAptEVafqBQE8IYBo9uZAMAwJdEcA0uiJNPxDoCQFMoycXkmFAoCsCmEZXpOkHAj0hgGn05EIyDAh0RQDT6Io0/UCgJwQwjZ5cSIYBga4IYBpdkaYfCPSEAKbRkwvJMCDQFQFMoyvS9AOBnhDANHpyIRkGBLoigGl0RZp+INATAphGTy4kw4BAVwQWZXQ0n9GGJhCAAASeJ9AX05iFcZDjeL54s8AxGuks5D3P60l06YhBAAJJAphGEg0VEIBARADTiKgQgwAEkgQwjSQaKiAAgYgAphFRIQYBCCQJYBpJNFRAAAIRAUwjokIMAhBIEsA0kmiogAAEIgKYRkSFGAQgkCSAaSTRUAEBCEQEMI2ICjEIQCBJANNIoqECAhCICGAaERViEIBAkgCmkURDBQQgEBHANCIqxCAAgSQBTCOJhgoIQCAigGlEVIhBAAJJAphGEg0VEIBARADTiKgQgwAEkgQwjSQaKiAAgYgAphFRIQYBCCQJYBpJNFRAAAIRAUwjokIMAhBIEsA0kmiogAAEIgKYRkSFGAQgkCSAaSTRUAEBCEQEMI2ICjEIQCBJANNIoqECAhCICGAaERViEIBAkgCmkURDBQQgEBHANCIqxCAAgSQBTCOJhgoIQCAigGlEVIhBAAJJAphGEg0VEIBARADTiKgQgwAEkgQwjSQaKiAAgYjAJE1jRSX0nSipjNjuarNBRrtJNvm8Op+X9p5kEvQ9EQJ7qdfHpD0m0nvLnY7DNFZRjndKj0r+knjbekC6XvqqtLRULa/UzsXSXBFcos9HJB//oHRmEX+RPn0u1zl+dhG/TJ9nSZsX++P8sJkdI90o3SC5zzc27GAFtf9ew2OaNB8lx1vUkTnXtW2TBDLaLjTHVXXub0pXSX+VfA18P7w1o89xNPF3wves791rpD9LH8w8sb8Lv5V2k16SecxCmi2U7fLq7OuSv3tXSNdJ5rznQpIYdIxvrpxyrBo9UWm4SNs7Sz7+F7UTnK59t6+WTbTztGRDsFmUxQP1xVujEvPmjtI/JH9Bc0rOOJzzhdI50jKS9w+W/i75hsgth6ihTa3pTKOLHP1l8HWpa7XMwbWd44+Uxx3SmkU+L9bn8ZLvrbeNMcfUqQ5Uxd+k1xUNPqzPp6QtUgdU4ja7L0ieBZtT05lG22ydl79jH6nkvEOR6+cqsUGbOTk+O/icUjeN8pg/aOM/0quKwJv16XOuE5z00KJuv0rd0dqOnNBPhDnpS8F5olDOOLYr+q+++iyr2EOSzSOnLFajuySboPts8nrSRY5+eo9S2s7RprFvLUHPTN2vZ4A5JSfH6DyvVvBJqX7Nfq+YH2bDSvmwa9M0RrlHfW/6+1QvtyjgB2VOyWKb1Ui9pUzjONX5HOX00m7sV46oeEp3s+T3wbWkraXzooZF7AR9eqqVU3LGcbJOdE9wsnMVM9iccqoa2cjWltxn/QYcdI4ucuzCNEbhuJQARa/NnmmcNghepS6HY3Qqzwx87Hq1Sr+uOJ47G2vTNEZhG43Zs+m7pZOiyiA2H12coN1IIX95PNPwlM9lK+mmYrv+8bgCvnB+utsQDi/26+3Kfa87bFq0T7VpEt9IjW8PDnDM0+WXBnXV0Oba2Vg6cki7UapHzdHX/DDpEsnX4XfSh0ZJKDh2lBw9ffb9Ui2ra8evi0tq8XHvOm+X+j1Q7pf14+63yflGYVvv5+UK+F7w69f365Wp/TZNw4uf/gXhPdJR0v1FEq/R532phBS/oGj/bn2eL5VmEx1yr4J2Si+ejaOsrJM8HJzIMfezUlBXhlxvk9tf+veAdqNWjZKj+35Q8uLeZtL6ktdevHj3WVeOqYyaYz0Nv297Buq1jTaL87Zp+eFVLeU94dekSZdxsfU98ID0PsmvPOabVcZtGn4azFW0i7a9MLRPJRuvbVQXTKNE7ezPSLtKr48aFLHyPOMyjQFdDa3aWS38C9Kvh7acbAPPzE6R/DS3uf1M8iug12y86Dht5R1KyA8fL3z7tXUcxa9A/gWi1LAZ5Dj6nLZzeM3N4/6pdJG0e26C4zYNLyKtUei1+vQN6llGdbppJ/dTOVXepIqdJK/qvkyKFm7KY8vzjOvJ7hmQ+6yXFRSYl+zMUVlOwQOl6gJu1G4csYXmOKhvL/K9QooWpwcdl6obV45eNPca0SekK1OdLSD+fh3zz4p+VZzDedtQ/HpcLb7+LuVs+X9ru90bF1tn7YfuMZIfIj+RbKJDy7hNY2iHauBFF3/JouJ8vHD6Rek30gnSNpJnLFEpz+NzjqP4p12vXdTLYgVukzyTiIrfM202p0tzhZYUDb9b7B9R7I/6sdAc3a8XmpcPEvCszqX6U3fQLDs0So5lJ354eEXfT0D/AjfOcqlO5tffUqXZO2+X+j3g6+9S1he7E/kYha2XDGyK9XKVAr431q1XLHTfT9ickvr1pH6sb4SL68Fi3+sBXpgpi53PP1/a4aNXkG8p7tlNzrQ6Zxzb61xu53f9sviVy+sAnr5Xixd4B82YXO9z7V07btBu2znuqs5PDhKwQf9Lqj9hg6bPjmlYGZWjDeNWyb+elcU3/JnDOi7qc3KMTlX+5LpXrdKLxZ6NVYsXEf2qHZUNFHQOe0SVA2I5eY/C9kD1/Y2g/x8W+ebMNHNyzLpJnEeuaXh9w1OserHLXS3Z8aplW+04UU9T68XTKi/i5ZScwdoELpTOlnyTuhwk2biqf9zl6bLP97XnmoT/tmUao+S4qzL1lHTLSsYei18fDwhH8f/Btjn6xjXvEySvE5X6jLbn/j+dMJKTY3iggv5i3SH59dplG8m/LmxR7Jcf12vDr6vRrLlN0xjl+ntsd0v+ha8sXhB/RPLDPKdksR3WyF+muaJjt/X2GQN698XwusY7K20u0baf5pZ/Ri3LVtqYk3xT+9y+mL6ILn4q3iulXl2KZs9/DBtH2dCzGxvgTdINkmHWHfi9ivkPvj4t1Yun+HPSnZL79CzJ+x7LsNJ2jp6tfVO6XPI0169cV0i7DUusUt92jiepL/cRaS4zz9wco9P5FdkPA197MzKr8p6rtj9XO9dK1Vnuhtqfk/wXxM7BD0fvT8s9upZyOUTyw9ljc/5+NfmKVH9YKxSWLLZZjcLTp4N+v/fTfJSyrw72wJfKPEkb48jsOrsZOWajGthwFjhGA5iFvLNyzGoUERgQszv7/fTgAW0GVX1AlX4SLB7UqFbXxjgadJ/VlByzMA1tNAsco0HMQt5ZOWY1iggMiXka+LEhbVLVXiDzT4RNSlvjaJLDsLbkOIxQXv0scIxGMgt5Z+WY1SgiMGWxWRgHOY7nppkFjtFIZyHvTv7vSQSHGAQgMKME/IpAgQAEIJBNANPIRkVDCEDABDAN7gMIQKARAUyjES4aQwACmAb3AAQg0IgAptEIF40hAAFMg3sAAhBoRADTaISLxhCAAKbBPQABCDQigGk0wkVjCEAA0+AegAAEGhHANBrhojEEIIBpcA9AAAKNCGAajXDRGAIQwDS4ByAAgUYEMI1GuGgMAQhgGtwDEIBAIwKYRiNcNIYABDAN7gEIQKARAUyjES4aQwACmAb3AAQg0IgAptEIF40hAAFMg3sAAhBoRADTaISLxhCAAKbBPQABCDQigGk0wkVjCEAA0+AegAAEGhHANBrhojEEIIBpcA9AAAKNCGAajXDRGAIQwDS4ByAAgUYEMI1GuGgMAQhgGtwDEIBAIwKYRiNcNIYABBZlIJjPaEMTCEAAAhCAAAQgAAEIQAACEIAABCAAAQhAAAIQgAAEIAABCEAAAhCAAAQgAAEIQAACEBgjgf8CHQclssnzvLkAAAAASUVORK5CYII=)
Answer:Given: A given table with values for X and P(X).
As we see from the table that P(X) = -0.1 for X = 3.
It is known that probability of any observation must always be positive that it can’t be negative.
Hence, the given table is not the probability distributions of a random variable.
Question 3.State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
![](data:image/png;base64,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)
Answer:Given: A given table with values for X and P(X).
As we know the sum of all the probabilities in a probability distribution of a random variable must be one.
![](data:image/png;base64,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)
Hence the sum of probabilities of given table = 0.6 + 0.1 + 0.2
= 0.9 ≠ 1
Hence, the given table is not the probability distributions of a random variable.
Question 4.State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
![](data:image/png;base64,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)
Answer:Given: A given table with values for X and P(X).
As we know the sum of all the probabilities in a probability distribution of a random variable must be one.
![](data:image/png;base64,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)
Hence the sum of probabilities of given table = 0.3 + 0.2 + 0.4 + 0.1 + 0.05
= 1.05 ≠ 1
Hence, the given table is not the probability distributions of a random variable.
Question 5.An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable?
Answer:Given: An urn containing 5 red and 2 black balls.
Let R represent red ball and B represent black ball.
Two balls are drawn randomly. Hence, the sample space of the experiment is S = {BB, BR, RB, RR}
X represents the number of black balls.
⇒ X(BB) = 2
X(BR) = 1
X(RB) = 1
X(RR) = 0
Therefore, X is a function on sample space whose range is {0, 1, 2}.
Thus, X is a random variable which can take the values 0, 1 or 2.
Question 6.Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?
Answer:Given: A coin is tossed 6 times.
X represents the difference between the number of heads and the number of tails.
⇒ X(6H, 0T) = |6-0| = 6
X(5H, 1T) = |5-1| = 4
X(4H, 2T) = |4-2| = 2
X(3H, 3T) = |3-3| = 0
X(2H, 4T) = |2-4| = 2
X(1H, 5T) = |1-5| = 4
X(0H, 6T) = |0-6| = 6
Therefore, X is a function on sample space whose range is {0, 2, 4, 6}.
Thus, X is a random variable which can take the values 0, 2, 4 or 6.
Question 7.Find the probability distribution of
number of heads in two tosses of a coin.
Answer:Given:
A coin is tossed twice. Hence, the sample space of the experiment is S = {HH, HT, TH, TT}
X represents the number of heads.
⇒ X(HH) = 2
X(HT) = 1
X(TH) = 1
X(TT) = 0
Therefore, X is a function on sample space whose range is {0, 1, 2}.
Thus, X is a random variable which can take the values 0, 1 or 2.
As we know,
P(HH) = P(HT) = P(TH) = P(TT) = 1/4
P(X = 0) = P(TT) = 1/4
P(X = 1) = P(HT) + P(TH) = 1/4 + 1/4 = 1/2
P(X = 2) = P(HH) = 1/4
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Question 8.Find the probability distribution of
number of tails in the simultaneous tosses of three coins.
Answer:three coins are tossed simultaneously. Hence, the sample space of the experiment is S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}
X represents the number of tails.
As we see, X is a function on sample space whose range is {0, 1, 2, 3}.
Thus, X is a random variable which can take the values 0, 1, 2 or 3.
P(X = 0) = P(HHH) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAgCAMAAAAynjhNAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOmaQOma2OpDbZgAAZjoAZrbbZrb/kDoAkNv/tmYAtmY625A625Bm27Zm2////9uQ/9u2//+2///bcIk3dwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAeElEQVQ4T8WSSw6AIAxEBxVFUUT8YO9/UcW4LRAT4mxn+lpagCKipYlw11bHbMD9Z5Otd350J271RVaWD7VhiEfVnFv1FqTjX+BpasGE10J07MnI9PADe7FzmADL/jYyavchwij0VmzvU48gw8IPGbWxydjDMjd6AbB/A910BVULAAAAAElFTkSuQmCC)
P(X = 1) = P(HHT) + P(HTH) + P(THH) ![](data:image/png;base64,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)
P(X = 2) = P(HTT) + P(THT) + P(TTH) ![](data:image/png;base64,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)
P(X = 3) = P(TTT) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAgCAMAAAAynjhNAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOmaQOma2OpDbZgAAZjoAZrbbZrb/kDoAkNv/tmYAtmY625A625Bm27Zm2////9uQ/9u2//+2///bcIk3dwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAeElEQVQ4T8WSSw6AIAxEBxVFUUT8YO9/UcW4LRAT4mxn+lpagCKipYlw11bHbMD9Z5Otd350J271RVaWD7VhiEfVnFv1FqTjX+BpasGE10J07MnI9PADe7FzmADL/jYyavchwij0VmzvU48gw8IPGbWxydjDMjd6AbB/A910BVULAAAAAElFTkSuQmCC)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Question 9.Find the probability distribution of
number of heads in four tosses of a coin.
Answer:Four tosses of a coin. Hence, the sample space of the experiment is S = {HHHH, HHHT, HHTH, HTHH, HTTH, HTHT, HHTT, HTTT, THHH, TTHH, THTH, THHT, THTT, TTHT, TTTH, TTTT}
X represents the number of heads.
As we see, X is a function on sample space whose range is {0, 1, 2, 3, 4}.
Thus, X is a random variable which can take the values 0, 1, 2, 3 or 4.
P(X = 0) = P(TTTT) ![](data:image/png;base64,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)
P(X = 1) = P(HTTT) + P(TTTH) + P(THTT) + P(TTHT) ![](data:image/png;base64,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)
P(X = 2) = P(HHTT) + P(THHT) + P(TTHH) + P(THTH) + P(HTHT) + P(HTTH) ![](data:image/png;base64,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)
P(X = 3) = P(THHH) + P(HHHT) + P(HTHH) + P(HHTH) ![](data:image/png;base64,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)
P(X = 4) = P(HHHH) ![](data:image/png;base64,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)
Hence, the required probability distribution is,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARwAAADFCAYAAACGsk2zAAAACXBIWXMAAAsTAAALEwEAmpwYAAASuklEQVR4Xu2bCawlRRWGGRV1EFcEAZEZVERx3DEKogQS4xoFESRiDAJGUYkLBI3GBY0hSNSIiTuKRuMCcQGXiEGGJSoqexgExXkgakBAhACCwPP/sXtsevrOO3eqb3fd7q+Sf2539amqc77qd7p6mU02oUAAAhCAAAQgAAEIQAACEIAABCAAAQhAAAIQgAAEIAABCEAAAhCAAAQgAAEIQAACqQSWBTpYDNhgAgEIQKAVAkNJOEOJw5M6D7HgYyt/fnMx19FIF+8XtcQOAhCAQCoBEk4qQdpDAAJhAiScMCoMIQCBVAIknFSCtIcABMIESDhhVBhCAAKpBEg4qQRpDwEIhAmQcMKoMIQABFIJkHBSCdIeAhAIEyDhhFFhCAEIpBIg4aQSpD0EIBAmQMIJo8IQAhBIJUDCSSVIewhAIEyAhBNGhSEEIJBKgISTSpD2EIBAmAAJJ4wKQwhAIJUACSeVIO0hAIEwARJOGBWGEIBAKgESTipB2kMAAmECJJwwKgwhAIFUAiScVIK0hwAEwgRIOGFUGEIAAqkESDipBGkPAQiECZBwwqgwhAAEUgmQcFIJ0h4CEAgTIOGEUWEIAQikEiDhpBKkPQQgECZAwgmjwhACEEglQMJJJUh7CEAgTICEE0aFIQQgkEqAhJNKkPYQgECYAAknjApDCEAglQAJJ5Ug7SEAgTABEk4YFYYQgEAqARJOKkHaQwACYQIknDAqDCEAgVQCJJxUgrSHAATCBEg4YVRza7hCnp8tXTO3EeD4tAS+owaL0qppG+Zgb8c3VFbr4C2S7W6STimM769fn+Q+5vqfFPV9/SwVR+nXI7TxZely6bLC7x37cnrCuNFYDlD7BelKqeuEE/HRfxCfl8z5EmmN9Dlpa6mLEvHxCXLkk9L5ha7Q7y+lPbtwUGNEfKy6slvRJseEE4olYvRcBXmXdK7kRFOWzbVxsbSyUtfXZiSOZXLuLOmn0oMk7x8r/VXasi/HG8aNxGL/V0vbSydLOSacC+WXWT+siNGMz5PWSj53Zl0iHN8hJ/4mPbFwxuf3p6V/S0+btYPqP+Jj6YbP199IpxbtclvhhGIJGSnA44ogj6hMwhe1/ZbKfp+bkTj2aZioB6vuX5ITTy4lEotPvvKWOeeE88wa1H217/gO7AB2hKPPicNqvmxX+HhUJj6WbrxeG159HVr4N+iEs1xB/lG6TfIy9MXSaVIuJXJyfVPOXtfg8M9U96eG+r6qIrFUfcs14TywAeCuqnN8XlnMukzLsfRn58LHg2ftYDFOZBhfGNdKz5KyTThtPjS+vQjUgX9N+lSxH4GVi83T5YgnrV5c93jpIfUD7CcRuLOh9ZOKujMbjuVQtYOcOF7yrfe3cnCo8MF3FmZ2QUY+redKmwnHnTvgL0gvlM6Qrl5vxLwrHi33bm5w0XW+RXlUwzGq2iNgxm+W/Ifsh8g5Fb848HOwP0s+H/aX7sjEQT9kP1x6fyb+THSj7YTjgbwauFs6SHrcxJE5AIH1CbxLVX5L+Pb1D/Ve48cFfnazhXSt5IS4S+9e/c+Bj+vHF3o/3J77Ms197k6K1m8eXi25nd9A5FIicfiNmp/y14tf1d4j5XJLFYmlGkOuz3CqPnrFUP5R1/nPan9ajqUfflPlZ3q/mpVjlX6X8vEZsvWdxGaVNtk+w4nwWirgsg+vls6R/B2Ay1clt31jsd/3TyQOHhq3N0sR3uVofjPl7578Cr/LEvHRL0N8q1cvP1CFX43Puizl45FywC86Fiq6Xttu5885XP86KYeyVCz3+hgykp1fEfoDqbJ4aewl3g3SYyr1fW1G4niNnLPdUytO+nsWf7g4b6/Fq5xzXuGYeT3Z+E3Vh6sBzGg7ck74Irp7w/i/V10XtzARH+vuDX6F8xRFfJHkq0G17K0dAzupVt/HbmTifCXz2wd/FV2+sj1G2z6x5u3DvyrjXBOOk43fbvph5xsq+oS2T6wGMKPtyDnhhONzYquKD35A67a5fYdTujjohPNrRekVgOUrVVn20saC5GcfnpyrpJetO9r9RuTksldemX1F8ifs/uTez6HKV7Xde908YjSWE9R8QbpV8pfg3vYn+l2UiI9+9mC7Jp3YgZMRH726sS+XSr6orpGcgPbrwD8PEfGxdOUF2liQ6rdU1ec7HbndOEwolpBRY/d5VQ4lDlOdh1jwsZ3zfx44RiNdnMVr8ejg2EEAAiMjQMIZ2YQTLgT6JEDC6ZM+Y0NgZARIOCObcMKFQJ8ESDh90mdsCIyMAAlnZBNOuBDokwAJp0/6jA2BkREg4YxswgkXAn0SIOH0SZ+xITAyAiSckU044UKgTwIknD7pMzYERkaAhDOyCSdcCPRJgITTJ33GhsDICJBwRjbhhAuBPgmQcPqkz9gQGBkBEs7IJpxwIdAnARJOn/QZGwIjI0DCGdmEEy4E+iRAwumTPmNDYGQESDgjm3DChUCfBEg4fdJnbAiMjAAJZ2QTTrgQ6JMACadP+owNgZERIOGMbMIJFwJ9EiDh9EmfsSEwMgIknJFNOOFCoE8CJJw+6TM2BEZGgIQzsgknXAj0SYCE0yd9xobAyAiQcEY24YQLgT4JkHD6pM/YEBgZARLOyCaccCHQJwESTp/0GRsCIyNAwhnZhBMuBPokQMLpkz5jQ2BkBEg4I5twwoVAnwRIOH3SZ2wIjIwACWdkE064EOiTwLLA4IsBG0wgAAEItEJgKAlnKHF4UuchFnxs5c9vLuY6Gukit1RRVNhBAALJBEg4yQjpAAIQiBIg4URJYQcBCCQTIOEkI6QDCEAgSoCEEyWFHQQgkEyAhJOMkA4gAIEoARJOlBR2EIBAMgESTjJCOoAABKIESDhRUthBAALJBEg4yQjpAAIQiBIg4URJYQcBCCQTIOEkI6QDCEAgSoCEEyWFHQQgkEyAhJOMkA4gAIEoARJOlBR2EIBAMgESTjJCOoAABKIESDhRUthBAALJBEg4yQjpAAIQiBIg4URJYQcBCCQTIOEkI6QDCEAgSoCEEyWFHQQgkEyAhJOMkA4gAIEoARJOlBR2EIBAMgESTjJCOoAABKIESDhRUthBAALJBEg4yQjpAAIQiBIg4URJYQcBCCQTIOEkI6QDCEAgSoCEEyWFHQQgkEyAhJOMkA4gAIEoARJOlBR2EIBAMgESTjJCOoAABKIESDhRUthBAALJBEg4yQjpAAIQiBIg4URJYQcBCCQTIOEkI6QDCEAgSqDPhPMIOfmRqKM1u0O0v2oj20aarZDR2dI1EeMGm7er7jbp0IZjXVelxmJ/nyrdJZ3ctfMZjTcPHFN9nPl520bC2VInhf8wb5UWi23v3yitkd4rPVCqli20c460UFSu1u8tktvfJJ1S1N9fv+7Lx1z/k6L+XP2eKu1e7Lf5c4A6O1PaNtDpg2XjuPcqbM3ix9LB0vIl2m+t41+SLpQukdZK35Y2W6LdNIdTYqmO8ynteC7aLk9Qh5+Uzi90hX5/Ke05YaA6b+8fLfk8u1i6TDJTs22z5M7Rsab4mNV56yQQKV+R0b8rhsu0/QbJ7b9e6+D72rd9tTxXO76KOplUT+7Nte+TaeV9rDfZ5HXa/7v0sFr9pN1IHA9S49XS9pKv5kutcF4uGyfCTYtBP6jfwySvvjzepBWO/yCcYN4imZPL86T/SI8u9jf000Us5fiO8Tzpr9I0K5yIj+9Qn3+TnlgM5nn/tOTz6GlFXfWnzvuLOmj+Ty6MfCG7SPqdVHJt6GZdVcTH1HOiHGyWHFN9zOm8vfcPJ1LqCadsc7o27pG2Kip8crjPJzV0elxx7IjKMZ9U/sOsF6/OFqR31g9M2I/E4ZO0XPVFEs7nZP+9ynhlolwq4TgB/7DBz5eorr4abDALzUlqLB73AdKl0h6Sk2/bCWcf9ekEXS3bacdzdVSt3rtV3o7vX1L9Yva2ov2ODe3rVV2cEx5z1hxT57qz87aNW6r6JNb3F1RhII8tDuynX18tvXyulw+p4k/SxyQvt18s7SA56dSLk9gZkvtrq/gEdL/R8goZ+haqLHcHGj5UNl6dlbeN1SY/186dgT4iJqmxeAwnes/TmZEBN8LmB2rz+Vq7csV6fUN/Vd6Oz7z9x1wt5X5kLhqGWK9qHjim+hhh1cp520XC8XLZf8RXF1Pp5x1NycaHb5d8G+J7869Jfnbg/Unlch3YTbJ918VLfl+NfzblwM+WvZfAjvUEyc9vHIefPWwzZV9tmTfF8nB1/gGpaaXR1rj1fnxxOV46S/pW7WCTj0fL5lXSroXtCv36Nu0b0p9r7bvYbfKxD44birXJxw3Zl8daOW9nmXB8a/BW6UXSF6QbCs/9R9V09SoD89XU9i+UvIIpE1V5vPr7D+149fSYpoMzrnul+v+tZB+mKY8rjL1q+4X0DMmx7iydI0WfSU0z5lK2TbF8UI18u/jHpRq3cNy3P75lc5K4WdpfuqPWb5OPn5HN+6TTJD8LulLyKvGQWtuudpt87JJjJM4mHyPtWjlv2044vnIvVPRGbfse/fBKRH6WU3243BTsWlXeLR0klYE22ZX99JFw6rdTTf411ZWrMV/FvyN59Xed9G7p8dKbmhrNuK4ei29nD5Q+OuNxy+6d1Lxa9EPfayWv+napjV330Yc/Kznh+KK2rbS95Cvx6VLkWZjMWi11H7vmGAmm7mOkjW1aOW/bTji+Kq0s5BPAtzterfiPqix+E+VVyaSykw74ZN9X8n2jVwKTStmP3+50WfyH8XypfE0/zdi3FMZ+HV4t3ve9uN/WdVmaYvmEHDhWurFLR4rxfEvkVY5vrcrS5ONzdNC2x0gXFIZe5bxHcgLqepXT5GNfHEtu9d8mH+s2k/ZbOW/bTjiTnK3W+wq22QRD+3OC5DcNP5L8HOdlkldKTaXsx312WezT36V60oj48IfCqM7eycaq10f6TLFpimUPdeg/3IWKtta2bV33a6mNslyd1C8+XtleInmlUpYmH1cVB+vPA8v9p1fad7HZ5GNXHKPxNfkYbdvKeVt/wh8dPMXuL2o86RboSB07V/pVMYBP+pdK/jbD9+bXFvXlj/8I7pSmfY5S62bqXd8Hb8zqxgNdLF0l1f8gnqI6J5vfTe1NWoOmWJq+BfIzlt9Ir00b7j6t/QzLt0R+dlUtvo2urq6afPRtqMuK+7T8/379XKmZtb7b5GNXHKPBNPkYbdvZeeurbqRM+g6n3tbPc66vV2rff3D+aGt57dje2rcPJzW08TOQ6mvpBpN1VdE4ygYna8N/ZPXiJP1PyZM3qazSAY836Q2bH4r61tLJ1MUx/1S6Uoo8NO4ylsLFmXyH40RzluTnemXx+eH4yrdjk3j7Gc0a6XLJz25cNpf81tDf56z8X9UG/x0Kx2qQc3/eLjUp/ix6QfI9nm297e8rJhU/2/Efm5+BlMVLdH8xavkEKste2liQ/AzIfXtl4GWhix9ieWUz6XarMFv3s1QcpaFv6RakWyX76e3zy4P63UPyK+2m20K/crS9vzPyeE6s3m/y0UnHt2SO1zbfkLaRIqWLWEo/nOgXJLMwE2+/vzy4gd+Ij7ur/YnSpZIvNk4gTkDVb6s2xNvnnt9UuZ2vwJdJ35WeLEVKxEf3k3JOlH7MkmOqj1mdt9FJiUxwaeNbpI29JSn7eJc2fJJFbwvbiuO4FnyfhlWT7TzEgo9NMzd93TxwjEYViiVkFB2xsNtUv6dIx07ZrjR/iTZ8NdthivZtxeGr6WFTjDsL03mIBR/bmfl54BiNNBRLyCg6YsXOD0g39gGk/8vDI6ccc1ZxTOlGK+bzEAs+tjLV996aD6WEYgkZzQGRocRh1PMQCz6280cxDxyjkS52/c1H1DHsIACBARIg4QxwUgkJArkSIOHkOjP4BYEBEiDhDHBSCQkCuRIg4eQ6M/gFgQESIOEMcFIJCQK5EiDh5Doz+AWBARIg4QxwUgkJArkSIOHkOjP4BYEBEiDhDHBSCQkCuRIg4eQ6M/gFgQESIOEMcFIJCQK5EiDh5Doz+AWBARIg4QxwUgkJArkSIOHkOjP4BYEBEiDhDHBSCQkCuRIg4eQ6M/gFgQESIOEMcFIJCQK5EiDh5Doz+AWBARIg4QxwUgkJArkSIOHkOjP4BYEBEiDhDHBSCQkCuRIg4eQ6M/gFgQESIOEMcFIJCQK5EiDh5Doz+AWBARIg4QxwUgkJArkSIOHkOjP4BYEBEiDhDHBSCQkCuRIg4eQ6M/gFgQESIOEMcFIJCQK5EiDh5Doz+AWBARIg4QxwUgkJArkSIOHkOjP4BYEBEiDhDHBSCQkCuRIg4eQ6M/gFgQESWBaIaTFggwkEIAABCEAAAhCAAAQgAAEIQAACEIAABCAAAQhAAAIQgAAEIAABCEAAAhCAAAQgAAEIQAACHRD4L6b9Ve06Q5fqAAAAAElFTkSuQmCC)
Question 10.Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
(i) number greater than 4
(ii) six appears on at least one die
Answer:Given: A die is tossed two times.
When a die is tossed two times then the number of observations will be (6 × 6) = 36.
Now, let X is a random variable which represents the success.
(i) Here success is given as the number greater than 4.
Now
P(X = 0) = P(number ≤ 4 in both tosses) ![](data:image/png;base64,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)
P(X = 1) = P(number ≤ 4 in first toss and number ≥ 4 in second case) + P(number ≥ 4 in first toss and number ≤ 4 in second case) is:
![](data:image/png;base64,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)
P(X = 2) = P(number ≥ 4 in both tosses) ![](data:image/png;base64,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)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
(ii) Here success is given as six appears on at least one die.
Now
P(X = 0) = P(six does not appear on any of die) ![](data:image/png;base64,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)
P(X = 1) = P(six appears atleast once of the die) ![](data:image/png;base64,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)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Question 11.From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
Answer:Given: A lot of 30 bulbs which include 6 defectives.
Then number of non-defective bulbs = 30 – 6 = 24
As 4 bulbs are drawn at random with replacement.
Let X denotes the number of defective bulbs from the selected bulbs.
Clearly, X can take the value of 0, 1, 2, 3 or 4.
P(X = 0) = P(4 are non defective and 0 defective) ![](data:image/png;base64,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)
P(X = 1) = P(3 are non defective and 1 defective) ![](data:image/png;base64,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)
P(X = 2) = P(2 are non defective and 2 defective) ![](data:image/png;base64,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)
P(X = 3) = P(1 are non defective and 3 defective) ![](data:image/png;base64,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)
P(X = 4) = P(0 are non defective and 4 defective) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAI0AAAAmCAMAAADOW+i6AAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmZmZmaQZrbbZrb/kDoAkDo6kGZmkGaQkLbbkNv/tmYAtmY6tmZmtrbbttv/tv//25A625Bm27Zm27a229uQ29u229vb2////7Zm/9uQ/9u2//+2///bUQWCigAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACf0lEQVRYR+1Y2VbCMBBtXauoFIpbLSpFJeDSlOT/v81MlqYtaZsAIudonopnlps7kzuJnve/dspAHjzsNF9rMjo+3yM0KE72Bw0O6e+gWT6+qZrhMBOf5HZGk/hJ/tph/+S9ic6GLmb8B/JhHe8cDYl4e9DpqYBRIHCplPLemMSUo5hfXQs0NBnKkB/BoeCpexXe3abtFvmZrBMSaDxsjaEcWHlvCEdQAyWSH5ocl8jbQaMFt4iH1iFnO2h06iLeWhNhO2hSdYRoeqKkJlJ9bF8q7d3hgw5KclK3JZHsFq4vAgVNnGWm5N1xZoI2NHlg4CFdp3GseCTRXRsa7BsmYyubVlmbjNIh3h80+DJrRWOkAZkIc+Ik5VMOVpkKMpp47miM5TMncMHIZZWhQY3XJsWN3ImcDdbcFBRUPhogYmXE0CyvGW1x3fCHuriZSFGpPBhkNF05zvZoNq+UmMkcTaG5VXaMU+Dn9IbNZkBD6movRdisfs5abN3MrHcYnCY0NFEXCh3Q9DfrdFaGK2iUl6GCRr6sstgaif0i6OJqKys1grknlapVoFYSLm76bPZ/shMbQgYI0w0K+bG3hOaZ9yoHS5FWkqSGhjfmoNMB3EPI6NlbwGaala3qPg98/yj28JATpJd8I+gw8g3RvUGwKF2y+AXbFo2ITu4zNHwpPwVkcqAYqC4/YboB0aR/04NKqTbQYbqd5aOten0pHlBCe5yoyYPw61Mcy4XYC+isDZDCuD6F1KuBTzW3FwNMZfEeRAoMi+CAZtWUjsOMjidsUw8e+3aJRRPmCtxg5obBHcK4RDDYvs/oVMzU5atbKJjF7B8JJIL5zNDIMG5B/pL1N4PpOImZ84t4AAAAAElFTkSuQmCC)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Question 12.A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.
Answer:Given: head is 3 times as likely to occur as tail.
Now, let the probability of getting a tail in the biased coin be x.
⇒ P(T) = x
And P(H) = 3x
For a biased coin, P(T) + P(H) = 1
⇒ x + 3x = 1
⇒ 4x = 1
⇒ x = 1/4
Hence, P(T) = 1/4 and P(H) = 3/4
As the coin is tossed twice, so the sample space is {HH, HT, TH, TT}
Let X be a random variable representing the number of tails.
Clearly, X can take the value of 0, 1 or 2.
P(X = 0) = P(no tail) = P(H) × P(H) ![](data:image/png;base64,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)
P(X = 1) = P(one tail) = P(HT) × P(TH) ![](data:image/png;base64,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)
P(X = 2) = P(two tail) = P(T) × P(T) ![](data:image/png;base64,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)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Question 13.A random variable X has the following probability distribution:
![](data:image/png;base64,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)
Determine
(i) k (ii) P(X < 3)
(iii) P(X > 6) (iv) P(0 < X < 3)
Answer:Given: A random variable X with its probability distribution.
(i) As we know the sum of all the probabilities in a probability distribution of a random variable must be one.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXsAAAA8CAMAAACwyupZAAAAAXNSR0IArs4c6QAAALFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmZmOmaQOma2OpDbZgAAZjoAZjo6ZjpmZjqQZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kJBmkLbbkNvbkNv/tmYAtmY6tpA6tpBmttv/tv+2tv//25A625Bm27Zm27aQ29uQ29v/2/+22////7Zm/9uQ/9u2//+2///baakUnAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFU0lEQVR4Xu1bDVfbNhS1PEoxHduAQNdBsq1rcVY2Q7rGTvT/f9je04f1YclWUiNzEukcODixde+7er5+kkyWpZYUSAokBV6ZAjU5/VLk81fG6kjoVPmnrDxdH0m0ryvMCnTHn9TiK5C0j6+5REzaT6Z9U5Az+LmejEACTgokBZICSYGkwIEqUBNHS2V+lNGmC2IVN7QkSfso2mfbGckfDKjtLGkfR/sMXOeHfw2stKwTSfosqwg5M8DqlPexxO9afizkhOOw/CRKNAXA8pPNRFPbAupY/lREjhAXLT/tF0408N0qf2wilT2NGBvA3x9Y6nSJVQ9jj2T5q1tznqYJUltTuLG074GUEG38AULANfQjIT89Kn4aAl3dEPLGG6MjphDIMSx/c2NP08bTfvmbcy+5F1KiN4XI+xAhMro4XW8W7XzTQCjJfbaZWXPR786iESy/uXys/LS+M+/p8vzqv06U/ZAd7YNUYkTbUTIRSpyFVsM20gKZftfckPzOQQIs3xKuJvnnvwtyFcSYn6Rpj4MJTPET2Iw8Xdf556VYtxMUkNdTAZFqlHow6ZfiUnMCyUqHfH5HkC7vmBOHvsnPIu/DjL9EGbYzNde3E2rw7jHw5T2HXIr7bKUdK2HhCnNtAWK4eMyed9nE1WnivQs3MCYJrlJgZ7TEnGopNMXF/er8waDUh0mfizef7ExQkHRxtpYIFw88PbFv+qdM1NZ8etKJ0Qbt1YzH1p4NTm+DwAQ+EFDPeVwoA5KuS5e2ymyAORejle2av/38NGhW8O32/Q0mP+zBs87Yr5ZCU7C9eYOSD1MQeH53cm+SsZRhsYpf0DtbF1QGYiSdOwwerx61C2FIe0YCo9O0387wA+dKZVPYI8I577KsadBE2PoaXj6hv6OD8pSfQ0pJCpyXSWkQE0dUb17tsV/edxt/eN77taeL4Zc5GI6FjR+wnHXcNY785jpgcKFvrBlC4O21fADNG3ReTXtJQWqvUxrA7M17urotWC3fxr639l6/93iGeR/4tPeNGrNis6kcNLUP9BwYtX9+XW9n8wqH0Mp79hzU8l4i+zDx+wC/l3Yj8q5Xe08Y4lmrdDISqjwLeG/Srb132FyFk/Be58PBbXimAdT5X2i5b//AZ1OrvaLAtTcp+TEH65wWQeU979vj9x7PZkrotYweFEvCesh1NO2xI3kT1TA7oE93RhHFUtDRH7wYvs6e4H5gf4Q0bsZSze0MDytePqlnraQgA2yP+WluTHd9j5bY3q8Yw+YDjifDYiHh69WbW1nnDFaHyABLHFGj8azTgmrOISBaXosQKU7BcKyEfDJwhQ/lNZm3BgaFbw5zRN3QzJJKaVyTHwtyglVdkN/TjwBETu5UJrPHNM9uqGDB+LkdCwp4xEZcHDNgH6Z7XttCsmth/nD1bUbmImDsnS5JfvW14OIL+KEkgpksuWS1DmhvBiVsakB7DX8IC753l6wqTYK0b3HA4gMw3afsi7k3oP9CM4y+I9nHPoF7Zsn76rC0SvBddNkXcxeMwHPNMPqOZId7BF57pq5T6DAFZuBYvMRppvUrBGWPTQGPrChtCswogblB2MNabyGzvwn5HhK0PUnnlXhqERTomn1nMCKwOEqI7gpaUwyukR6lUqMHjTscnZa0H11nV4dqPUkbgaR9DO2daR+6eBODYMJICiQFDkuB3RbJDiv2qaNxaU+XO+yWTB3AYeE//YKvFqT20gqIDSmxgy63d6qk/UsLj/07/T5pH0P6pH0Uld0gPO+T50wxBM46p3wb9lbCFIQPB9O1IQVv4tv/5384AadIkgJJgRdX4H85c6xEM8LeIQAAAABJRU5ErkJggg==)
Hence the sum of probabilities of given table:
⇒ 0 + k + 2k + 2k + 3k + k2 + 2k2 + 7K2 + k = 1
⇒ 10K2 + 9k = 1
⇒ 10K2 + 9k – 1 = 0
⇒ (10K-1)(k + 1) = 0
![](data:image/png;base64,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)
It is known that probability of any observation must always be positive that it can’t be negative.
So ![](data:image/png;base64,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)
(ii) P(X < 3) = ?
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
= 0 + k + 2k
= 3k
![](data:image/png;base64,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)
(iii) P(X > 6) = ?
P(X > 6) = P(X = 7)
= 7K2 + k
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(iv) P(0 < X < 3) = ?
P(0 < X < 3) = P(X = 1) + P(X = 2)
= k + 2k
= 3k
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPwAAAAqCAMAAACgEsZnAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjo6ZjpmZmY6ZmaQZpDbZrb/kDoAkGY6kLbbkNv/tmYAtmY6tpBmttv/tv//25A627Zm27aQ29u229v/2////7Zm/9uQ/9u2//+2///bjrvaZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADbElEQVRoQ+1Zf3ubIBCGdFncjzbpknVbTefWmnSLyvf/eDtAFBQ84OnUtOGf5Hm8g3uP4+VeJeQyLhk4vwwcv/4MC5odPlG6DHQKW2Ik63JHr57C1soWD6RMF+ePvlg/5sHgV5CsnN6HpWye1sHgBYwsNGWvB3y5Xz/OE01gVOE7X20pXf8NXGae5uHgAccxebtlT8iJfpnnXoZFFbXzpEjeJHiWctinV3LVBbYrjPc3Zfr+/BmP7RNK6bsfISel5D4bB3b2+yawYQxZmndX871kWXr7vyui+Dyj3pKlUDuUXvOuR/IB/O4pvcHboBz87oE+KH6GNDVVfPCRFSPJr2oLzb68+E/y9mfACGVq7wTKOy30g7gws5WtWgxDoqupHM8VTDqO/GKpUjryH+SAc6j1Pjjulg/aIWc8Q8VHC0t0DCFBrZry2nrdgZDqTq5R1r+BROM2l5B511Mk8jgK1SMKwhjskIjDoY0iWbG0d4Ythu287eHCETTyi6VLjr7abnCnIIt258WOy6rnC5nVyX4lFqLOF9+68dgNYdcaNSV2FR+6/BIx9fcDnwSxEOCPCUxugJcpUKPc080fy0TVtkMNbsNWTWX6zJngWz7MhqUjvyCc552VXHpRuWa05UHQ/YJf/HWnLGGb4LOlnfzB2eyTXYawcqOmDPADW2PIL1ipyVmTL+0PVgR2H0VzNdG1Za8Xp2tD89vOaxGXIY9NqSlf8Ib80sFjQP2f98ArwjO31HqUi+sn4DxzLdeZJ42aMs78UJFq8kuUvc8dCXztOEj2slfRK7YXL/j6V12fxAXT918H2gw1NVVtcUXZlV+K8F66/2w5VNUAJ3rzyKukda5v0a4A5/X5oNcQaGrK557vyK86mhene853qpzqDo/AC3HXiy6tceOesIdQZTZhaXZ4upry6vBM+VV+l1te1L/+h9rfUvX2/h4Rlj4b7zGt+ljjqUPEjMM+Z6Pqmo81Qzqkk0LU50z0fPuxxq1DutUT4+NRgZOY1B2ZQ4fYQ4rxmQQctqgEYtchLt8YHyyOSZ7rQOz3cj+sGJ9JwGGLxgCJ8cHimOS5WcJ+IjnGZxJw2KImeeENM58vxgeLY5LnNRCHDhlk+yCfScBhi+ayp3brEMsEMT5YHOM/1z7WDOgQM64Yn/GRXVYcJQP/AGlPYHYNWgmQAAAAAElFTkSuQmCC)
Question 14.The random variable X has a probability distribution P(X) of the following form, where k is some number:
![](data:image/png;base64,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)
(a) Determine the value of k.
(b) Find P (X < 2), P (X ≤ 2), P(X ≥ 2).
Answer:Given: A random variable X with its probability distribution.
(a) As we know the sum of all the probabilities in a probability distribution of a random variable must be one.
![](data:image/png;base64,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)
Hence the sum of probabilities of given table:
⇒ k + 2k + 3k + 0 = 1
⇒ 6k = 1
![](data:image/png;base64,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)
(b) (i) P(X < 2) = ?
P(X < 2) = P(X = 0) + P(X = 1)
= k + 2k
= 3k
![](data:image/png;base64,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)
(ii) P(X ≤ 2) = ?
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
= k + 2k + 3k
= 6k
![](data:image/png;base64,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)
(iii) P(X ≥ 2) = ?
P(X ≥ 2) = P(X = 2) + P(X > 2)
= 3k + 0
= 3k
![](data:image/png;base64,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)
Question 15.Find the mean number of heads in three tosses of a fair coin.
Answer:Given: A coin is tossed three times.
three coins are tossed simultaneously. Hence, the sample space of the experiment is S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}
X represents the number of heads.
As we see, X is a function on sample space whose range is {0, 1, 2, 3}.
Thus, X is a random variable which can take the values 0, 1, 2 or 3.
P(X = 0) = P(TTT) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAgCAMAAAAynjhNAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOmaQOma2OpDbZgAAZjoAZrbbZrb/kDoAkNv/tmYAtmY625A625Bm27Zm2////9uQ/9u2//+2///bcIk3dwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAeElEQVQ4T8WSSw6AIAxEBxVFUUT8YO9/UcW4LRAT4mxn+lpagCKipYlw11bHbMD9Z5Otd350J271RVaWD7VhiEfVnFv1FqTjX+BpasGE10J07MnI9PADe7FzmADL/jYyavchwij0VmzvU48gw8IPGbWxydjDMjd6AbB/A910BVULAAAAAElFTkSuQmCC)
P(X = 1) = P(TTH) + P(THT) + P(HTT) ![](data:image/png;base64,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)
P(X = 2) = P(THH) + P(HTH) + P(HHT) ![](data:image/png;base64,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)
P(X = 3) = P(HHH) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAgCAMAAAAynjhNAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOmaQOma2OpDbZgAAZjoAZrbbZrb/kDoAkNv/tmYAtmY625A625Bm27Zm2////9uQ/9u2//+2///bcIk3dwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAeElEQVQ4T8WSSw6AIAxEBxVFUUT8YO9/UcW4LRAT4mxn+lpagCKipYlw11bHbMD9Z5Otd350J271RVaWD7VhiEfVnFv1FqTjX+BpasGE10J07MnI9PADe7FzmADL/jYyavchwij0VmzvU48gw8IPGbWxydjDMjd6AbB/A910BVULAAAAAElFTkSuQmCC)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Therefore mean μ is:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKYAAAA8CAMAAADbqzFgAAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjo6OjpmOjqQOmY6OmZmOmaQOma2OpC2OpDbZgAAZjoAZjo6ZjpmZmYAZmY6ZpDbZrb/kDoAkDo6kLbbkNvbkNv/tmYAtmY6tpA6tpBmtpCQtrbbttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2/9vb//+2///bzSROdwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADBElEQVRoQ+1Yi1bUMBBtquJGQUUoPituAIWC1mrz/7/mzKRJkzZsm+5uCdqcs7Dd00xu7tx5JEmyjIWBWRgo2cEVTz/PstY2ixTpeSIOfm5jYo65BUDET+RjgblDB1WcreBzukOTi6mFgf+BgZJ5RnzpU+asE95SsPhgJnXG0rWjmzqLEGYCbn9y4+CMs7QXjK0cmGWMbCZ9ecaZO/ryjBMnyDNKP3fZ6slzj3TKq1dWxBbH1+PXQnnOdMCQ+YnTe1eHAQvvU57IAQ3EI3PsFcF78KS0Vr1w0/ZGcvcpzzrDhPeLI8xS5eg7Kn1ihcQGHW32KE+ZU14WALP5Cv8BbvWSIAfRGS7PEots0algPpdpbAiJKKUvK5mr70oHYwfIs1s0UTvAsrJWsvTrJWdvjb0wmHfEKM2hUaQftSnhFsEBwCDPzvsCjOo9I86j60ZUCnefTWG6QrudUTHUgVlnhhSnh/CbsKFfdE9sMgc2jUdKpBV/auaEsVkSzFYjgF1bDmt1QC5dviv+2nIyeb+1GQZTmbacfiI0nUEwLZ5atIVyHuUMYpP4UCkkwOlm/wZmdXRjaHG0OeR0FGJ31O8/0JYtmLj1e2H61W8lJB3ppPhCBWedBUR6M8dZSH5Zq9SsHKa0aT2fJj/epetvQxdPygYh1NNpoxBFWNBD8mbF+1uSAn4rQelwq0hOh7+3wLl6BKfz9NPvs2dDtU4XS/IWVaGmwwX/UrIY3535zj9gLAU28UjXOP2Qs6fnWgPJpCa/n8tDyDRR51eXpU0TUbo8+2fc++s2HZJPmPZCfpiemBuBeXq/ifrbOLww/3wfAWqHr1jR7LdacbjxhmvFprrT4+yDmip7mDZmdiibFmwqTftKW9EiwtkXZg93BGj7DUfFR7S6MyM35yn7/jA+mG1DYuGMDqaXzMdxBTKz7P6l5QK6qIfctg+mvAg6kT4Q/Ns3Z7HBbBpxbH71ZRS2vbHB9Hf4C8yJQlYh9CidLsXzoTPtRFKmTvM14niVO3TqmLreMs/PwF9Tx069PCyNCQAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 16.Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Answer:Given: A die is thrown two times.
When a die is tossed two times then the number of observations will be (6 × 6) = 36.
Now, let X is a random variable which represents the success and is given as six appears on at least one die.
Now
P(X = 0) = P(six does not appear on any of die) ![](data:image/png;base64,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)
P(X = 1) = P(six appears atleast once of the die) ![](data:image/png;base64,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)
P(X = 2) = P(six does appear on both of die) ![](data:image/png;base64,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)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Therefore Expectation of X E(X):
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 17.Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).
Answer:Given: first six positive integers.
Two numbers can be selected at random (without replacement) from the first six positive integer in 6 × 5 = 30 ways.
X denote the larger of the two numbers obtained. Hence, X can take any value of 2, 3, 4, 5 or 6.
For X = 2, the possible observations are (1, 2) and (2, 1)
![](data:image/png;base64,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)
For X = 3, the possible observations are (1, 3), (3,1), (2,3) and (3, 2).
![](data:image/png;base64,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)
For X = 4, the possible observations are (1, 4), (4, 1), (2,4), (4,2), (3,4) and (4,3).
![](data:image/png;base64,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)
For X = 5, the possible observations are (1, 5), (5, 1), (2,5), (5,2), (3,5), (5,3) (5, 4) and (4,5).
![](data:image/png;base64,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)
For X = 6, the possible observations are (1, 6), (6, 1), (2,6), (6,2), (3,6), (6,3) (6, 4), (4,6), (5,6) and (6,5).
![](data:image/png;base64,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)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Therefore E(X) is:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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Question 18.Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.
Answer:Given: Two fair dice are rolled
When two fair dice are rolled then number of observations will be 6 × 6 = 36.
X denote the sum of the numbers obtained when two fair dice are rolled. Hence, X can take any value of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12.
For X = 2, the possible observations are (1, 1).
![](data:image/png;base64,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)
For X = 3, the possible observations are (1,2) and (2,1)
![](data:image/png;base64,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)
For X = 4, the possible observations are (1,3), (2,2) and (3,1).
![](data:image/png;base64,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)
For X = 5, the possible observations are (1, 4), (4, 1), (2,3) and (3,2)
![](data:image/png;base64,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)
For X = 6, the possible observations are (1, 5), (5, 1), (2,4), (4,2) and (3,3).
![](data:image/png;base64,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)
For X = 7, the possible observations are (1, 6), (6, 1), (2,5), (5,2),(3,4) and (4,3).
![](data:image/png;base64,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)
For X = 8, the possible observations are (2,6), (6,2),(3,5), (5,3) and (4,4).
![](data:image/png;base64,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)
For X = 9, the possible observations are (5, 4), (4, 5), (3,6) and (6,3)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJkAAAAqCAMAAACX3AlTAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmY6ZmZmZpDbZrbbZrb/kDoAkGY6kLaQkLbbkNv/tmYAtmY6tpA6tpBmttv/tv//25A627Zm27aQ29v/2////7Zm/9uQ/9u2//+2///b5TPDYAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACiElEQVRYR+1Y7VbbMAx1UrZ6bKWFDNjWFGoYhNA48fs/3SQ5H+4pYLmlJucM/UoTOb6WdK+UCvFp44xAka4DgJVXId4BL9511TIAWX2ZTB4P2o6/uMl+8ZHp+UMRDZm6qPjI4MDRkFVnm3Eia36uxSiRmfxCILJiyS7MSNmsktZGh4wiNcpsjhxZWA8I82aX70uOUGtsQTMrCYV58pu9ofl75rSMYv7AXnlkR5Ofb9wt9A8+1d4dmsmJ7jOMDomSKODnUkBqvm6E/hZtHtg9WZNNhSglttfK9tinBAGqKYavAHgfZSYHZBCoJYQMrzB0gFCfEsrIQdOUu9YsngripGVbVVpOTW6vbYLjmSvWQ8yGu0V6vWjRKBvH41qTdc3NFR5CVkqopwFZk/VTnXILTfUvYAtX8JIi7fSAyJkuoNCH5g/3uiRuIXsxcsNxhytfiN/wdJF1+XKyea66oPmR+VAwnr+RTVrdI9OzRyCBfeNWnQWnBtaHFcAuAxBEx03iJcoIWJNF5eaWapDStgphr0hdgQQoLZH1zM08EqCjH/UAvAFxgiwg/z6yBzgwd2WVFTLz9D1JvsAx0MwdCpDfyktYwx9l9ps1VHor6twqXCUXz35YkAtYY3oBYKzYaz4j+lrGVPIPY5eOWHHoRecf2OTBZ2XA5KzEs876mlO9ovG3slLjNxut4yNDCZ9j2avJDRQ2djqfqclamPteFXzuBzyn0RMEB6b1UjLE2ayAmje9lB6ws3cpDnhtdtitVnPO4N3Z54C7tMjYf1gc+4vT6jMVvxUobszYTPZF5bXnBkW2JgGACWUDHzg8gpaSQ5V9UdG6Gr+37S4ae84t521azlh+nHf9bz7/AJvpOUwNBsePAAAAAElFTkSuQmCC)
For X = 10, the possible observations are (5,5), (4,6) and (6,4).
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAqCAMAAACawnkUAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjo6ZmY6ZpDbZrbbZrb/kDoAkGY6kLaQkLbbkNv/tmYAtmY6tpBmttv/tv//25A627Zm27aQ29v/2////7Zm/9uQ/9u2//+2///bW6FHCgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAClElEQVRYR+1Y6VLjMAy2A0vDcrSl7AJh2XCkgTj1+7/eSrYTK7NMI3kmaYfBv0JqRR86Ph1KfZ/jtYDdXmj9468UYP1LLCJV4e+X2ZNqi0ymrb3VJ29p+qRS5QIkKv0gkTPL12oufN6IYm1z4msfl68S8+Hd+fDtNlovP44XHyCr86P2r1KNvhEacD7/IjCTHys+WyCyRsYvM+aHRWpuizNpglRCRheGT7zePuZar2TwLMro0zu+UvtyTepNJeczvqqUm7ZYDyxgLkX1KkXlmIwtwMRaXyHz+yCv4M8HCHUNwWTOZQV/TFvC77sNVHbPrI2n161jsXKBpqzEEZ8AYa+ILbrOwz+hGQGn+emwHsSAxnkzHI8Kmd/kIdpMvrCFf/Yun/s0hHWi/eLbKvu9CphcXzfPwQ6jOxGgw1fnEGcR327TF/qSBmD5ifwI9gQR/GKVdczhEjhDZo21Gt51bh3g+xRL/Kfj05jBR2Uovs6DxL/rvhMexzeGhf37Hv+6b/T4zNUbpIj/7iD+EpyVIEKQuAzt7Nflr8vdMOrsNgfI3wG/OH4OXOKfHCdDiiAJHYT/aERgenQp6uoHvgCbgV8wxw9fPwjY/8mYZb7B6sA+I1VxTlgd2PqWvXhI61/o6qDJV+8ccKpfHZT6XrWRb/cLJ/V/ZHXQ5PcsdCquDlIWDzwdg1uOMGOuMb5ARzf54MJQQK6E1YFIDcUnX4xI8PWrg/LkDwQ7cwoh+PrOSaJVdNc1uEBNMB/UzCGYlvvpiwA2kNbPl8yS3eOLFUxkEtllXB0EfMydRX/NTxKTHbI68GEutJ8rUc10HiarA+h6PmC84g2lYXXgSpQtp8OnyOrAYK16Yrgqrg5CAzYhPgacr3DlH7pOOST4RCHSAAAAAElFTkSuQmCC)
For X = 11, the possible observations are (6,5) and (5,6)
![](data:image/png;base64,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)
For X = 12, the possible observations are (6, 6).
![](data:image/png;base64,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)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Therefore E(X) is:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ E(X) = 7
And E(X2) is:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ E(X2) = 54.833
Then Variance, Var(X) = E(X2) – (E(X))2
= 54.833 – (7)2
= 54.833 – 49 = 5.833
And Standard deviation ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFgAAAAcCAMAAADSgoh8AAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZmY6ZrbbZrb/kDoAkDo6kDpmkNv/tmYAtmY6tpBmttu2ttv/tv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///bMRBcpAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABwElEQVRIS+1UXVfCMAxtgVXlQ5wKbkpRieBW1v//90zStRucHTp2fPLYl22Q3N7c5EaI//OHFQA55MQFsa+beNCQCDMvhqTFc8okGmPf57smCBaf0QwK0JnNpRxtBD3GLYCQbfNlIagTmSilnBRmmvVArp4QSyOuQORJlyo2fyCcvaSHTjDE3PRoC0tcIhkhqrSTSenKsDk+zS2/QyeD0zKAeFQpCc0Idn8n5T3dJbMvxQrVTTAqsbm7ug/lbSOCxjsQprCsjFGzlwPWbJSvA0ZrvNGxZ1nC/LOSp4clxtrwr+rR/81YRrG0ogxZVRp6q6OjVA8boZRBuBrYMW2AcWzcXUg2KjK4bBoIUgLfDitFgxUkgEATltq/M/AlKYKfYfzBopDGDtNrGxib2Q77F2V8XBHX4GejppzDMJ3APBHAc0njfN4s/w2LZ6om+Bn14xQS+7gmxm62W5JQPPaP/FylXuwOfE7UwRK+HVuc4u9UZgaVdqLzHJPf8RNl5ZIuWY/6Zd+6dsMZi9p57V8vOw9nt9fKrHdFCzhiPPQx+zl6eLu1cWPbDUbTHluKBL5yH7csGmV9XQBEHX8d3u9E/wBGCygLFvj2YQAAAABJRU5ErkJggg==)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFIAAAAaCAMAAAAT+fuoAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgBmZjo6ZjqQZpDbZrbbZrb/kDoAkDo6kDpmkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bYQAmjQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABa0lEQVRIS+1U21KDMBDdUJVqVby0VGq0iBaI5P+/z70kBGYql5lOn5qH9kA2Z3fP2QBwWWdVIFdz1pTS7PZ9SticmOalnBM+Jba6mRI1K0ZvfPjXiRSwb3uAOiaHIk9ZLPHhFTNZQlf0OqDxeus7lLKO21LxRB6tAQr1CKCjHZiUUgU0TpnjyT6lTVldfV2CJpQrzBfQEOXvgXY/qK1elZ7S26YXKA2vgI7ymkzRmeaJ4vuNV4oad8qabPUpBAEheXs7WgMAfg6cU0aojm9jsYHXN7oloU2i1IrnNqCBzusYZczZF7tdOxskAW2QgsLvGw/oP1abLvbd28hMxJ+iMe6XnivynldARxvniA2PkFtNIoa4/9YLn8qVP9A3F1JIAVomqVslDZFN6Q1mxlCPBhlJr6UIptEMgzpIhbl6KHnULU25IR0CGqFsEqe8ecahoGGRpvka7hCYDL2/J20CGivz9F+hkYyX7Y4Cf0gQHJMVNQXMAAAAAElFTkSuQmCC)
⇒ Standard deviation = 2.415
Question 19.A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X.
Answer:Given: The class of 15 students with their ages.
Form the given information we can draw a table:
![](data:image/png;base64,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)
P(X = 14) ![](data:image/png;base64,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)
P(X = 15) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACUAAAAgCAMAAACijUGCAAAAAXNSR0IArs4c6QAAAEJQTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOma2OpDbZjoAZrbbZrb/kDoAkDo6kNv/tmYAtmY625A625Bm2////9uQ///b7M5aaQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAfElEQVQ4T82TRxKAIAxFY+8UNfe/qiDIjhBHB82KxUt+ygcgU6Cq00prOzAoAP1PCmW1JYfUhYk+iX0KSNvjGaW42UjIdA9u9gNFrkR+znveXjy+x8vzeqIbdJ5nUkaxI8qF/7M3hGqgcI771XkeFwFELe95VGYR46t3OABotgOdquEZJAAAAABJRU5ErkJggg==)
P(X = 16) ![](data:image/png;base64,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)
P(X = 17) ![](data:image/png;base64,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)
P(X = 18) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACUAAAAgCAMAAACijUGCAAAAAXNSR0IArs4c6QAAAEJQTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOma2OpDbZjoAZrbbZrb/kDoAkDo6kNv/tmYAtmY625A625Bm2////9uQ///b7M5aaQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAfElEQVQ4T82TRxKAIAxFY+8UNfe/qiDIjhBHB82KxUt+ygcgU6Cq00prOzAoAP1PCmW1JYfUhYk+iX0KSNvjGaW42UjIdA9u9gNFrkR+znveXjy+x8vzeqIbdJ5nUkaxI8qF/7M3hGqgcI771XkeFwFELe95VGYR46t3OABotgOdquEZJAAAAABJRU5ErkJggg==)
P(X = 19) ![](data:image/png;base64,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)
P(X = 20) ![](data:image/png;base64,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)
P(X = 21) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACUAAAAgCAMAAACijUGCAAAAAXNSR0IArs4c6QAAAEJQTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOma2OpDbZjoAZrbbZrb/kDoAkDo6kNv/tmYAtmY625A625Bm2////9uQ///b7M5aaQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAfElEQVQ4T82TRxKAIAxFY+8UNfe/qiDIjhBHB82KxUt+ygcgU6Cq00prOzAoAP1PCmW1JYfUhYk+iX0KSNvjGaW42UjIdA9u9gNFrkR+znveXjy+x8vzeqIbdJ5nUkaxI8qF/7M3hGqgcI771XkeFwFELe95VGYR46t3OABotgOdquEZJAAAAABJRU5ErkJggg==)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Therefore E(X) is:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ E(X) = 17.53
And E(X2) is:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ E(X2) = 312.2
Then Variance, Var(X) = E(X2) – (E(X))2
= 312.2 – (17.53)2
= 312.2 – 307.417 ≈ 4.78
And Standard deviation ![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ Standard deviation ≈ 2.19
Question 20.In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var (X).
Answer:Given: X = 0 if members oppose, and X = 1 if members are in favour.
P(X = 0) ![](data:image/png;base64,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)
P(X = 1) ![](data:image/png;base64,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)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Therefore E(X) is:
![](data:image/png;base64,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)
= 0 × 0.3 + 1 × 0.7
⇒ E(X) = 0.7
And E(X2) is:
![](data:image/png;base64,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)
= (0)2 × 0.3 + (1)2 × 0.7
⇒ E(X2) = 0.7
Then Variance, Var(X) = E(X2) – (E(X))2
= 0.7 – (0.7)2
= 0.7 – 0.49 = 0.21
Question 21.The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is
A. 1
B. 2
C. 5
D. 8/3
Answer:Given: A die having written 1 on three faces, 2 on two faces and 5 on one face.
Let X be the random variable representing a number on given die.
Then X can take any value of 1, 2 or 5.
The total numbers is six.
Now
P(X = 1) ![](data:image/png;base64,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)
P(X = 2) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAgCAMAAAAynjhNAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOpDbZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ2////9uQ/9u2//+2///bjLFCZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAcklEQVQ4T8WQyQ6AIBBDiwvuKwgqyv9/pku8MhITYq+dvnQKBJGdUoKri5qyAfWfbWWyuqsrdqoMMpk/VF4lbkXCN/UE3s+/wN+pAS90zqLGybf9iJmeyWTUitvQEuUliykbWHjniu+VgHHb0Jx6zHPRAzGTA60TjbGuAAAAAElFTkSuQmCC)
P(X = 5) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAgCAMAAAAynjhNAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADqQAGaQAGa2OgAAOjo6OmZmOmaQOma2OpC2OpDbZgA6ZjoAZjo6kDoAkNv/tmYAtmY625A625Bm27Zm29u229vb2////9uQ/9u2//+2///byGMlQAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAd0lEQVQ4T8VS2Q6AIAwb3uIBIt7u/39TNL5uEBNiX7d2TTuAKMAxZ3SnqufGAPa/MZpso61b4dBGiSxc1NwmHiRDKOsl+Ne/iPtVo27MqqYrw7Fh+vR8GupalbT4Xsh10eSvnp2rx6bUedRyQ5oNRy+E5LyHhHoBXJUEZzpYeoMAAAAASUVORK5CYII=)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Therefore Expectation of X E(X):
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ E(X) = 2
Hence, the correct answer is (B).
Question 22.Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then the value of E(X) is
A. ![](data:image/png;base64,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)
B.![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:Given: A deck of cards.
X be the number of aces obtained.
Hence, X can take value of 0, 1 or 2.
As we know, in a deck of 52 cards, 4 cards are aces. Therefore 48 cards are non- ace cards.
P(X = 0) = P(0 ace and 2 non ace cards) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
P(X = 1) = P(1 ace and 1 non ace cards) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
P(X = 2) = = P(2 ace and 0 non ace cards) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, the required probability distribution is,
![](data:image/png;base64,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)
Therefore Expectation of X E(X):
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEIAAAAqCAMAAAAwLX3tAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkGY6kLaQkLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29v/2////7Zm/9uQ/9u2//+2///bdXjrxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABrElEQVRIS+1Wa0/CMBRtQaUKyHzhECqOuXXr//+B3le3mUjWJmgg8X5YCmnPzj23OWdKXUT58lHr6x1y9Rut53thXUzov5iyeq2abPoBCPlN1eS4gnImAWIGBwr9qlSNh2pcKdVmz/EQ9E46aJFAmyGksisCTCg8jX0gBD7reZUI4QyQYAh6tg87biu6fL7CeXQQ9BsgCpIlpnxO7YdGZqCMVDSEnUEDKCHLiYxQ4YRGCtIPDvJkeahJEO4WZPMWIHAY3A1W0u2kxpF+A1d9IQigR0InMZL/7zk/BWy49Scedgc7sjiZIr/VyMkIngNQ+cQ5driTSBuEG/z9boJ9HCMLBsHhZSdb1eRoEH24gXGZ5edIn26xLwSiizTbrQBhHSOUQPT2y76JBiq5NorSQzSbRQh19vLOiEdAAkSb9bYJqY4ubqdv4KVL8dLjOINGSiPfBSHc9H2lSiOxEgUBzGn7t3BTdmyqaiin4zdKuEmiDDf8zIR3MHXWrws30jSGBeWNx1vVUJD14eYMZO0h5OMRLfzGgNtcvUCO4YrEF1tAVg6/47aj9+KPNnwBJJckgAp29GgAAAAASUVORK5CYII=)
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Hence, the correct answer is (D).
State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
Answer:
Given: A given table with values for X and P(X).
As we know the sum of all the probabilities in a probability distribution of a random variable must be one.
Hence the sum of probabilities of given table = 0.4 + 0.4 + 0.2
= 1
Hence, the given table is the probability distributions of a random variable.
Question 2.
State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
Answer:
Given: A given table with values for X and P(X).
As we see from the table that P(X) = -0.1 for X = 3.
It is known that probability of any observation must always be positive that it can’t be negative.
Hence, the given table is not the probability distributions of a random variable.
Question 3.
State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
Answer:
Given: A given table with values for X and P(X).
As we know the sum of all the probabilities in a probability distribution of a random variable must be one.
Hence the sum of probabilities of given table = 0.6 + 0.1 + 0.2
= 0.9 ≠ 1
Hence, the given table is not the probability distributions of a random variable.
Question 4.
State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
Answer:
Given: A given table with values for X and P(X).
As we know the sum of all the probabilities in a probability distribution of a random variable must be one.
Hence the sum of probabilities of given table = 0.3 + 0.2 + 0.4 + 0.1 + 0.05
= 1.05 ≠ 1
Hence, the given table is not the probability distributions of a random variable.
Question 5.
An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable?
Answer:
Given: An urn containing 5 red and 2 black balls.
Let R represent red ball and B represent black ball.
Two balls are drawn randomly. Hence, the sample space of the experiment is S = {BB, BR, RB, RR}
X represents the number of black balls.
⇒ X(BB) = 2
X(BR) = 1
X(RB) = 1
X(RR) = 0
Therefore, X is a function on sample space whose range is {0, 1, 2}.
Thus, X is a random variable which can take the values 0, 1 or 2.
Question 6.
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?
Answer:
Given: A coin is tossed 6 times.
X represents the difference between the number of heads and the number of tails.
⇒ X(6H, 0T) = |6-0| = 6
X(5H, 1T) = |5-1| = 4
X(4H, 2T) = |4-2| = 2
X(3H, 3T) = |3-3| = 0
X(2H, 4T) = |2-4| = 2
X(1H, 5T) = |1-5| = 4
X(0H, 6T) = |0-6| = 6
Therefore, X is a function on sample space whose range is {0, 2, 4, 6}.
Thus, X is a random variable which can take the values 0, 2, 4 or 6.
Question 7.
Find the probability distribution of
number of heads in two tosses of a coin.
Answer:
Given:
A coin is tossed twice. Hence, the sample space of the experiment is S = {HH, HT, TH, TT}
X represents the number of heads.
⇒ X(HH) = 2
X(HT) = 1
X(TH) = 1
X(TT) = 0
Therefore, X is a function on sample space whose range is {0, 1, 2}.
Thus, X is a random variable which can take the values 0, 1 or 2.
As we know,
P(HH) = P(HT) = P(TH) = P(TT) = 1/4
P(X = 0) = P(TT) = 1/4
P(X = 1) = P(HT) + P(TH) = 1/4 + 1/4 = 1/2
P(X = 2) = P(HH) = 1/4
Hence, the required probability distribution is,
Question 8.
Find the probability distribution of
number of tails in the simultaneous tosses of three coins.
Answer:
three coins are tossed simultaneously. Hence, the sample space of the experiment is S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}
X represents the number of tails.
As we see, X is a function on sample space whose range is {0, 1, 2, 3}.
Thus, X is a random variable which can take the values 0, 1, 2 or 3.
P(X = 0) = P(HHH)
P(X = 1) = P(HHT) + P(HTH) + P(THH)
P(X = 2) = P(HTT) + P(THT) + P(TTH)
P(X = 3) = P(TTT)
Hence, the required probability distribution is,
Question 9.
Find the probability distribution of
number of heads in four tosses of a coin.
Answer:
Four tosses of a coin. Hence, the sample space of the experiment is S = {HHHH, HHHT, HHTH, HTHH, HTTH, HTHT, HHTT, HTTT, THHH, TTHH, THTH, THHT, THTT, TTHT, TTTH, TTTT}
X represents the number of heads.
As we see, X is a function on sample space whose range is {0, 1, 2, 3, 4}.
Thus, X is a random variable which can take the values 0, 1, 2, 3 or 4.
P(X = 0) = P(TTTT)
P(X = 1) = P(HTTT) + P(TTTH) + P(THTT) + P(TTHT)
P(X = 2) = P(HHTT) + P(THHT) + P(TTHH) + P(THTH) + P(HTHT) + P(HTTH)
P(X = 3) = P(THHH) + P(HHHT) + P(HTHH) + P(HHTH)
P(X = 4) = P(HHHH)
Hence, the required probability distribution is,
Question 10.
Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
(i) number greater than 4
(ii) six appears on at least one die
Answer:
Given: A die is tossed two times.
When a die is tossed two times then the number of observations will be (6 × 6) = 36.
Now, let X is a random variable which represents the success.
(i) Here success is given as the number greater than 4.
Now
P(X = 0) = P(number ≤ 4 in both tosses)
P(X = 1) = P(number ≤ 4 in first toss and number ≥ 4 in second case) + P(number ≥ 4 in first toss and number ≤ 4 in second case) is:
P(X = 2) = P(number ≥ 4 in both tosses)
Hence, the required probability distribution is,
(ii) Here success is given as six appears on at least one die.
Now
P(X = 0) = P(six does not appear on any of die)
P(X = 1) = P(six appears atleast once of the die)
Hence, the required probability distribution is,
Question 11.
From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
Answer:
Given: A lot of 30 bulbs which include 6 defectives.
Then number of non-defective bulbs = 30 – 6 = 24
As 4 bulbs are drawn at random with replacement.
Let X denotes the number of defective bulbs from the selected bulbs.
Clearly, X can take the value of 0, 1, 2, 3 or 4.
P(X = 0) = P(4 are non defective and 0 defective)
P(X = 1) = P(3 are non defective and 1 defective)
P(X = 2) = P(2 are non defective and 2 defective)
P(X = 3) = P(1 are non defective and 3 defective)
P(X = 4) = P(0 are non defective and 4 defective)
Hence, the required probability distribution is,
Question 12.
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.
Answer:
Given: head is 3 times as likely to occur as tail.
Now, let the probability of getting a tail in the biased coin be x.
⇒ P(T) = x
And P(H) = 3x
For a biased coin, P(T) + P(H) = 1
⇒ x + 3x = 1
⇒ 4x = 1
⇒ x = 1/4
Hence, P(T) = 1/4 and P(H) = 3/4
As the coin is tossed twice, so the sample space is {HH, HT, TH, TT}
Let X be a random variable representing the number of tails.
Clearly, X can take the value of 0, 1 or 2.
P(X = 0) = P(no tail) = P(H) × P(H)
P(X = 1) = P(one tail) = P(HT) × P(TH)
P(X = 2) = P(two tail) = P(T) × P(T)
Hence, the required probability distribution is,
Question 13.
A random variable X has the following probability distribution:
Determine
(i) k (ii) P(X < 3)
(iii) P(X > 6) (iv) P(0 < X < 3)
Answer:
Given: A random variable X with its probability distribution.
(i) As we know the sum of all the probabilities in a probability distribution of a random variable must be one.
Hence the sum of probabilities of given table:
⇒ 0 + k + 2k + 2k + 3k + k2 + 2k2 + 7K2 + k = 1
⇒ 10K2 + 9k = 1
⇒ 10K2 + 9k – 1 = 0
⇒ (10K-1)(k + 1) = 0
It is known that probability of any observation must always be positive that it can’t be negative.
So
(ii) P(X < 3) = ?
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
= 0 + k + 2k
= 3k
(iii) P(X > 6) = ?
P(X > 6) = P(X = 7)
= 7K2 + k
(iv) P(0 < X < 3) = ?
P(0 < X < 3) = P(X = 1) + P(X = 2)
= k + 2k
= 3k
Question 14.
The random variable X has a probability distribution P(X) of the following form, where k is some number:
(a) Determine the value of k.
(b) Find P (X < 2), P (X ≤ 2), P(X ≥ 2).
Answer:
Given: A random variable X with its probability distribution.
(a) As we know the sum of all the probabilities in a probability distribution of a random variable must be one.
Hence the sum of probabilities of given table:
⇒ k + 2k + 3k + 0 = 1
⇒ 6k = 1
(b) (i) P(X < 2) = ?
P(X < 2) = P(X = 0) + P(X = 1)
= k + 2k
= 3k
(ii) P(X ≤ 2) = ?
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
= k + 2k + 3k
= 6k
(iii) P(X ≥ 2) = ?
P(X ≥ 2) = P(X = 2) + P(X > 2)
= 3k + 0
= 3k
Question 15.
Find the mean number of heads in three tosses of a fair coin.
Answer:
Given: A coin is tossed three times.
three coins are tossed simultaneously. Hence, the sample space of the experiment is S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}
X represents the number of heads.
As we see, X is a function on sample space whose range is {0, 1, 2, 3}.
Thus, X is a random variable which can take the values 0, 1, 2 or 3.
P(X = 0) = P(TTT)
P(X = 1) = P(TTH) + P(THT) + P(HTT)
P(X = 2) = P(THH) + P(HTH) + P(HHT)
P(X = 3) = P(HHH)
Hence, the required probability distribution is,
Therefore mean μ is:
Question 16.
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Answer:
Given: A die is thrown two times.
When a die is tossed two times then the number of observations will be (6 × 6) = 36.
Now, let X is a random variable which represents the success and is given as six appears on at least one die.
Now
P(X = 0) = P(six does not appear on any of die)
P(X = 1) = P(six appears atleast once of the die)
P(X = 2) = P(six does appear on both of die)
Hence, the required probability distribution is,
Therefore Expectation of X E(X):
Question 17.
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).
Answer:
Given: first six positive integers.
Two numbers can be selected at random (without replacement) from the first six positive integer in 6 × 5 = 30 ways.
X denote the larger of the two numbers obtained. Hence, X can take any value of 2, 3, 4, 5 or 6.
For X = 2, the possible observations are (1, 2) and (2, 1)
For X = 3, the possible observations are (1, 3), (3,1), (2,3) and (3, 2).
For X = 4, the possible observations are (1, 4), (4, 1), (2,4), (4,2), (3,4) and (4,3).
For X = 5, the possible observations are (1, 5), (5, 1), (2,5), (5,2), (3,5), (5,3) (5, 4) and (4,5).
For X = 6, the possible observations are (1, 6), (6, 1), (2,6), (6,2), (3,6), (6,3) (6, 4), (4,6), (5,6) and (6,5).
Hence, the required probability distribution is,
Therefore E(X) is:
Question 18.
Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.
Answer:
Given: Two fair dice are rolled
When two fair dice are rolled then number of observations will be 6 × 6 = 36.
X denote the sum of the numbers obtained when two fair dice are rolled. Hence, X can take any value of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12.
For X = 2, the possible observations are (1, 1).
For X = 3, the possible observations are (1,2) and (2,1)
For X = 4, the possible observations are (1,3), (2,2) and (3,1).
For X = 5, the possible observations are (1, 4), (4, 1), (2,3) and (3,2)
For X = 6, the possible observations are (1, 5), (5, 1), (2,4), (4,2) and (3,3).
For X = 7, the possible observations are (1, 6), (6, 1), (2,5), (5,2),(3,4) and (4,3).
For X = 8, the possible observations are (2,6), (6,2),(3,5), (5,3) and (4,4).
For X = 9, the possible observations are (5, 4), (4, 5), (3,6) and (6,3)
For X = 10, the possible observations are (5,5), (4,6) and (6,4).
For X = 11, the possible observations are (6,5) and (5,6)
For X = 12, the possible observations are (6, 6).
Hence, the required probability distribution is,
Therefore E(X) is:
⇒ E(X) = 7
And E(X2) is:
⇒ E(X2) = 54.833
Then Variance, Var(X) = E(X2) – (E(X))2
= 54.833 – (7)2
= 54.833 – 49 = 5.833
And Standard deviation
⇒ Standard deviation = 2.415
Question 19.
A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X.
Answer:
Given: The class of 15 students with their ages.
Form the given information we can draw a table:
P(X = 14)
P(X = 15)
P(X = 16)
P(X = 17)
P(X = 18)
P(X = 19)
P(X = 20)
P(X = 21)
Hence, the required probability distribution is,
Therefore E(X) is:
⇒ E(X) = 17.53
And E(X2) is:
⇒ E(X2) = 312.2
Then Variance, Var(X) = E(X2) – (E(X))2
= 312.2 – (17.53)2
= 312.2 – 307.417 ≈ 4.78
And Standard deviation
⇒ Standard deviation ≈ 2.19
Question 20.
In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var (X).
Answer:
Given: X = 0 if members oppose, and X = 1 if members are in favour.
P(X = 0)
P(X = 1)
Hence, the required probability distribution is,
Therefore E(X) is:
= 0 × 0.3 + 1 × 0.7
⇒ E(X) = 0.7
And E(X2) is:
= (0)2 × 0.3 + (1)2 × 0.7
⇒ E(X2) = 0.7
Then Variance, Var(X) = E(X2) – (E(X))2
= 0.7 – (0.7)2
= 0.7 – 0.49 = 0.21
Question 21.
The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is
A. 1
B. 2
C. 5
D. 8/3
Answer:
Given: A die having written 1 on three faces, 2 on two faces and 5 on one face.
Let X be the random variable representing a number on given die.
Then X can take any value of 1, 2 or 5.
The total numbers is six.
Now
P(X = 1)
P(X = 2)
P(X = 5)
Hence, the required probability distribution is,
Therefore Expectation of X E(X):
⇒ E(X) = 2
Hence, the correct answer is (B).
Question 22.
Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then the value of E(X) is
A.
B.
C.
D.
Answer:
Given: A deck of cards.
X be the number of aces obtained.
Hence, X can take value of 0, 1 or 2.
As we know, in a deck of 52 cards, 4 cards are aces. Therefore 48 cards are non- ace cards.
P(X = 0) = P(0 ace and 2 non ace cards)
P(X = 1) = P(1 ace and 1 non ace cards)
P(X = 2) = = P(2 ace and 0 non ace cards)
Hence, the required probability distribution is,
Therefore Expectation of X E(X):
Hence, the correct answer is (D).
Exercise 13.5
Question 1.A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes? (ii) at least 5 successes?
(iii) at most 5 successes?
Answer:We know that the repeated tosses of a dice are known as Bernouli trials.
Let the number of successes of getting an odd number in an experiment of 6 trials be x.
Probability of getting an odd number in a single throw of a dice(p) ![](data:image/png;base64,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)
Thus,![](data:image/png;base64,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)
Now, here x has a binomial distribution.
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
= 6Cx (1/2)6-x(1/2)x
= 6Cx (1/2)6
(i) Probability of getting 5 successes = P(X = 5)
= 6C5(1/2)6
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(ii) Probability of getting at least 5 successes = P(X ≥ 5)
= P(X = 5) + P(X = 6)
= 6C5(1/2)6 + 6C5 (1/2)6
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC4AAAAsCAMAAAAtr3oOAAAAAXNSR0IArs4c6QAAAGZQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjqQOma2OpC2OpDbZgAAZgA6ZjoAZmZmZrbbZrb/kDoAkLaQkLbbkNv/tmYAtmY6tv//25A627Zm27aQ29v/2////7Zm/9uQ//+2///b28k4uQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA4klEQVRIS+2UyxLCIAxFg1VjbX2hRVF5/f9PGlvRLhwDHbtwxiw6LE7hckMuwC+VFm3NTJpotSPO4f2bXLo4JbMAvq4yaNCTJgMPcp5BgxVZF1WpLrYaHGZdVI3pos1yMcfCP6u6CaMabvJzi95ikLPfEDPo4DF/CkeMk5cQBxbL60ONQ7YlFrdRuq83HO7rV76oip2qXr7YhWFxVezXQpQU1X7VAIcHKZYGLpQyQVLSEK4/pVmQrYeUYTY+sgQ8ZjsnBrr4ignJ4g7nBs4xfvmuOjJmeug6Rfq5Po35Gt/tfQO9Jws6rG7GOAAAAABJRU5ErkJggg==)
(iii) Probability of getting at most 5 successes = P(X ≤ 5)
We can also write it as: 1 – P(X>5)
= 1 – P(X = 6)
= 1 – 6C6 (1/2)6
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 2.A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.
Answer:We know that the repeated tosses of a pair of dice are known as Bernouli trials.
Let the number of times of getting doublets in an experiment of throwing two dice simultaneously four times be x.
![](data:image/png;base64,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![](data:image/png;base64,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)
![](data:image/png;base64,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)
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
= 4Cx (5/6)4-x(1/6)x
= 4Cx (54-x/66)
Hence, Probability of getting 2 successes = P(X = 2)
= 4C2(54-2/64)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 3.There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item?
Answer:Let there be x number of defective items in a sample of ten items drawn successively.
Now, as we can see that the drawing of the items is done with replacement. Thus, the trials are Bernoulli trials.
Now, probability of getting a defective item, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ we can say that x has a binomial distribution, where![](data:image/png;base64,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)
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
![](data:image/png;base64,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)
Probability of getting not more than one defective item = P(X ≤1)
= P(X = 0) + P(X = 1)
= 10C0 (19/20)10(1/20)0 +10C1 (19/20)9(1/20)1
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 4.Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is a spade?
Answer:Let the number of spade cards among the five drawn cards be x.
As we can observe that the drawing of cards is with replacement, thus, the trials will be Bernoulli trials.
Now, we know that in a deck of 52 cards there are total 13 spade cards.
Thus, Probability of drawing a spade from a deck of 52 cards ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Thus, x has a binomial distribution with ![](data:image/png;base64,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)
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
![](data:image/png;base64,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)
(i) Probability of drawing all five cards as spades = P(X = 5)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(ii) Probability of drawing three out five cards as spades = P(X = 3)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(iii) Probability of drawing all five cards as non-spades = P(X = 0)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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Question 5.The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one will fuse after 150 days of use.
Answer:Let us assume that the number of bulbs that will fuse after 150 days of use in an experiment of 5 trials be x.
As we can see that the trial is made with replacement, thus, the trials will be Bernoulli trials.
It is already mentioned in the question that, p = 0.05
Thus, q = 1 – p = 1 – 0.05 = 0.95
Here, we can clearly observe that x has a binomial representation with n = 5 and p = 0.05
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
= 5Cx(0.95)5-x(0.05)x
(i) Probability of no such bulb in a random drawing of 5 bulbs = P(X = 0)
= 5C0(0.95)5-0(0.05)0
= 1× 0.955
= (0.95)5
(ii) Probability of not more than one such bulb in a random drawing of 5 bulbs = P(X≤ 1)
= P(X = 0) + P(X = 1)
= 5C0(0.95)5-0(0.05)0+ 5C1(0.95)5-1(0.05)1
= 1× 0.955 + 5 × (0.95)4 × 0.05
= (0.95)4 (0.95 +0.25)
= (0.95)4 × 1.2
(iii) Probability of more than one such bulb in a random drawing of 5 bulbs = P(X>1)
= 1 – P(X ≤ 1)
= 1 – [(0.95)4 × 1.2]
(iv) Probability of at least one such bulb in a random drawing of 5 bulbs = P(X ≥ 1)
= 1 – P(X < 1)
= 1 – P(X = 0)
= 1 –(0.95)5
Question 6.A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
Answer:Let us assume that number of balls with digit marked as zero among the experiment of 4 balls drawn simultaneously be x.
As we can see that the balls are drawn with replacement, thus, the trial is a Bernoulli trial.
Probability of a ball drawn from the bag to be marked as digit![](data:image/png;base64,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)
It can be clearly observed that X has a binomial distribution with![](data:image/png;base64,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)
![](data:image/png;base64,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)
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
![](data:image/png;base64,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)
Probability of no ball marked with zero among the 4 balls = P(X = 0)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 7.In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers 'true'; if it falls tails, he answers 'false'. Find the probability that he answers at least 12 questions correctly.
Answer:Let us assume that the number of correctly answered questions out of twenty questions be x.
Since, ‘head’ on the coin shows the true answer and the ‘tail’ on the coin shows the false answers. Thus, the repeated tosses or the correctly answered questions are Bernoulli trails.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, it can be clearly observed that x has binomal distribution, where ![](data:image/png;base64,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)
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGUAAAAvCAMAAADNYfT3AAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkDo6kGY6kGZmkLbbkLb/kNv/tmYAtmY6tpA6ttv/tv//25A625Bm27Zm27aQ29u22////7Zm/7aQ/9uQ/9u2//+2///bqxRB+wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACLklEQVRYR+1X3VrCMAxtJ7ApyECFicoQC7g51vd/PJOu6/5ZK51X5Map/XKak+QkJeRmfQx8LymdwSG+oc5L3+G//j9dvJOjsyWEuVFyDz8HM/Se+mvCg/lgGBAGOE8XEEfo2kI5PdVpOWJarKKcl/Tuq3phhiDWGONve5JM90ygJMI1Wgxf8RrYmljJ/i5LbYZC5G8QAwUDlHRJndXVaWEysxKFB+DZuqlekCgkrufHBqQq0hxliOYo2jpHGSIYNokkIwol9W13esmjQsHatZGLwkeMepgZU5+lP/ah6Qm3SvnGg+4YZQqfeLqU6Qk3D9rI4YGJOPYLd+q3+jNKTF24TzD9nGk5sx3chAaNWRduRh8j/lmkGxURhappRSH0ZZ/UhTvxkB3+Wp4iV6M0hLvtgh2X7gBvBtYQbh60kG2KEuI0EFYhvoBP/ZaiNUXpTdNFFHVFKvxoM9ZA/R/GSNhCpXXGgAZUJ8uV3EIZLAnYlUfP2SaeqIWr+6VZDnwHujt6jkg8idhMCI0FhekuOyYlHtYhqZYcNW6cy4KRWnbCHB5k7yjlD+mKnH3Zt2bK34Xy88GD8T4bkNKx2GSY1E79KXahSw+ey0OaRVMZvnktGEzkXjEQByr7Sj5W7KSlfIFiU4KCy4aN/U2JFFuf2imHWGHzDVYV1hAbrAomdOVCMEQo+AYWJY01F4PSDfOyyN5FIj08hK/8laRXo/qn8MUnR+289OLTd3A72cbAL2zvM7i8DlryAAAAAElFTkSuQmCC)
Probability of at least 12 questions answered correctly = p(X ≥ 12)
= P(X = 12) +P(X = 13)+…+P(X = 20)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWgAAAAvCAMAAAD5EaCHAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkDo6kGY6kGZmkLbbkNv/tmYAtmY6tpA6ttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bgmczZgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEbElEQVR4Xu1bf1fTMBRNJyBTkY2pMFA7sYC0LN//4/nS/GiXtutNm7yj5yz/ODVw77tNX967yYQ4jZMCJwVgBf5ssuwzzZYP2eIb/FOTJjJCQfxY+exvfojnxU8hisvX6gP9mW4wQkFB8PNRAu/Xd0JuVxDDGZMYoSCWvHwK0nd/Q6s5v4TYzZjECAWxTMjn5YufH55Vik4hNCMUpConn7dN9u73IatC6ZwgdTBCQTKz8ZHfH0V19VjUQle1umqU9Km8o7RxEW8zZISCNObls9M7nRZamL/RSs5okND7Tba4hWiPT2KEGifTBMsRuirfNCeDJrckbqLBCAVFwMrH1chGaFH6uRrijExihELoCF4+rnKzQqcrmhmhIKFZ+TRNnxU62ZJmhIJ05uVTXLwaVk7o/VrvjrEHIxREnZVPS1QntCroIKZhkxihIGK8fEplG5mqw31s/SNEGZvECAURms0nyONzO+DDkqrmM22HVks0d4RgzYRCxAuhY8tZsoEnhU7eRIC9Kbd9WUJuQQ8pBGsmFKJzCB1yJGeFrvnAHt9+3StpSJJGsSJAIWKjdKjznR06NXn06rddtxc6LFlc9exwA0kiD+hZPKyUUIjQnKEL394ssutX+avZ9hrCpTIzuqPom6un+R26hxUTqoPVR9XjkzL0Drxvb1ZL9ZLI+57jqNlCe1hRoSYInTR0X+iOvXlkfQ7814D+3RXtY0WFChc6beie0B17U26HM+5MoX2suFDBQqcNPVe2cT0GEut+PdzozRTaf5niQgUL3UmikfmMbMYImntYWf3LBlJHlGeKQYkuVrGgjbtaXqsN+bw+o+DkA1Q8cd/no+cCcaG8Fd0Vuq8KOhAkKp/RxyryJqfInSpAbKdKBXd/uoE3Q/85d6CePtprTuFQ81NHK3QTc3PtagKfsVVdZtS/1OXd06cNCe061cEcEVBHH4L7UPLe+AFToCII7fjYmJtrV+GV7ZjOxJcOV03DYk/JjOkdjnb8SDEqVASh23yUMdG6dhUe+qjQQu7Inzr7qlpwdxypFvmgT3ekBR85u+1CvT3oI/QJ3T5yTozzUcZEy5eYwGdc6WaGEbruVGk4Z0Uqj+Lc9o4hptIguobKszMtdEooSAEVc0voxHx09LpTVYnbeoV5dive1qazgW3SowHal+dlWfspKaEQoeuYW6kjLR+Zv6cEYjtVJbnRtj4QLozFhBv/RyKsodQKqrTQCaEQnU3MrWtX6UJXStJYuU6VCB6cWtn9IcZRloYST0t3iz0ZFKKzjbl17YqXz8GJt90Do6ToTviMUIj0lEZWzby0odcvdGOE2Hc8/XUDl05SQUFKM4ZOfJpbJO6GUqo7YYxQkNDMfOy9KFdqpL8SxgAFKc0YemtJ55fmTDHVgm7eHgYoSGi7pJn4mNJa5eqStgekHYPC6JnECAVRZOajbofXD1fm9MleRIeYhk5ihIKo8fJR3y8wZuuq9dUKiGngJEYoiNm/xgcifZp0UuCkwP+hwF9Ff7RVdzff4QAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Question 8.Suppose X has a binomial distribution
. Show that X = 3 is the most likely outcome.
(Hint: P(X = 3) is the maximum among all P(xi), xi = 0,1,2,3,4,5,6)
Answer:As per the question,
X is any random variable whose binomial distribution is![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
![](data:image/png;base64,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)
![](data:image/png;base64,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)
It can be clearly observed that P(X = x) will be maximum if 6cx will bw maximum.
∴6cx = 6c6 = 1
6c1 = 6c5 = 6
6c2 = 6c4 = 15
6c3 = 20
Hence we can clearly see that 6c3 is maximum.
∴for x = 3, P(X = x) is maximum.
Hence, proved that the most likely outcome is x = 3.
Question 9.On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing ?
Answer:In this question, we have the repeated correct answer guessing form the given multiple choice questions are Bernouli trials
Let us now assume, X represents the number of correct answers by guessing in the multiple choice set
Now, probability of getting a correct answer, ![](data:image/png;base64,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)
Thus, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Clearly, we have X is a binomial distribution where n = 5 and ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, probability of guessing more than 4 correct answer ![](data:image/png;base64,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)
= P (X = 4) + P (X = 5)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 10.A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is 1/100. What is the probability that he will win a prize
(a) At least once
(b) Exactly once
(c) At least twice?
Answer:(a) In this question, let X represents the number of prizes winning in 50 lotteries and the trials are Bernoulli trials
Here clearly, we have X is a binomial distribution where n = 50 and ![](data:image/png;base64,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)
Thus, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAL0AAAAXCAMAAABpjr6KAAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZjoAZjo6ZjpmZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkLbbkLb/kNvbkNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/7aQ/9uQ/9u2//+2///bSGOzyAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACk0lEQVRYR+1WbVPbMAy2va71GGy0ZLDXMDYCjBSo58X//6dN8lvcxHacu44PO3zHAbEkP370SBYhL+uFgf+TgQfOriVf7gpvp+7e3ReaBmbt6a/5TkUeYrlr18Xg602p6d7h8vhbEoygyzvO0vve0Rt2FdULfdrFl2FkmxCiam114phT9Rk6aEdBaXHGCJFvrpPwW3ZFGhPr8QOl7DRFUGDog23fjlG4hHTVCkLyV1YtwvzxQPESzWpOGtr0VXHLbLd0s1O3LHXR3tCD//1D1a9HsrQJUTWgh6Amr+Y//A23kEezCiBDvgclOcZXl+Xot3ylmrEGbEIMXqG5hvRzq044RtXTSm1QXKAztDSiiy6Pvo2zLkFO6+8QJMJ9H1BQ9vOW0zV8cQnZ51746C37hGZTqwEHd83G5G28JKcr+DkzKY3t4xZQ4A3jcQTWJ4raJsRq5dE11B59Vw0OAprtCvlTNXBvObd1CfUSNQVEXRWtDe0orHgzfGkTPNHb6KbDXENtPWT4nhRCeIDk712OPPokgDj6rsKTvGjT8M0Fw2NcnRqfQDmbJprlUexeytPo48qZib5nuO8yA/Ty5N70B78ScuguPrpbet0nlUOwTEZrJvo09y59uhBdG81VLjQ//WJoUU9LzfS2Qcd0bW+yxVndB6S6sw1EpyP9tEDhTg0vqgE0tt3mHltHgKrZV4Kv1d7whS/sn/OSqgVY2yB9WLRhH9BvLX4EVCCA5KPoLgsWOJPo8aJkrFA3nNLF5x0Jhy91A+3+yT01uao95nRxlTbIPDk5AWHPyMw5Mdfh8FXcc7JC/gczZvS84fB1EPTkmeb70fDlhpUMtQUP2oRADrQ9Gr5gHJ8qW5gjpkwOhO5ZwvwF9C5MAtwfwn4AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
Hence, probability of winning in lottery atleast once ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGYAAAAXCAMAAADOf8o0AAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOpC2OpDbZgAAZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkGYAkGY6kLbbkNv/tmYAtmY6tpBmtrZmttv/tv//25A627Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bFAVF/wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABiElEQVRIS+1VXXOCMBAkKDb90Ba1LUWRatQ2VPL/f15zCVxCgCAPnel0eg/OaO5ud2+TMwj+4w9MQOTzw2gZbLH31nACMd1gkkgeP+WXIyWTg0gIeRqAPK22kFHcv3oTWbgNxJv80CGSqm8GP3GiMHXsOhh/LSUbdVrc1i064RikcVJz4VWVLsturBpxWobrj0aTYrFX9RBsZhi1kEQCpwgjkroxyGJzp/KS0+lL8zeE8copYxhShhoojpjN3l0UIHlOyYNxUoqoS/W4M+U1BPoAVaBD7NBobg4LWjdwRiBSuwPCNCfs1DAAjpCeBcOIbQyWnVc9aoLMY45zZrjxSd5Wc9nRaNPjjYbpHloZNymjGmmots2EON6F69ZtsobWr6Ywlqt+NUwZy6vgKO16N9YV8HmT4YPRrCtYfa/h5Q4GJrnirUp5AZqtdP8yIXIA8oVH9nbrGLtIKayqZ8XQvwaafHELDMpwE7xbwE3GnTYWZpQY2J3Wtrwea2hDtzr9yP/N9Xx/UeY34Z8fIec7HuAAAAAASUVORK5CYII=)
= 1 – P (X <1)
= 1 – P (X = 0)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(b) Probability of winning in lottery exactly once ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(c) Probability of winning in lottery atleast twice ![](data:image/png;base64,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)
= 1 – P (X < 2)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIYAAAAXCAMAAAD5ncdFAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjqQOmaQOpC2OpDbZgAAZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkGYAkGY6kLbbkNv/tmYAtmY6tpBmtrZmttv/tv//25A627Zm27aQ29u22////7Zm/9uQ/9u2/9vb//+2///b4FegEAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABnElEQVRIS+1Wa1PDIBAMfYm2GusDa7G2+CA1/P//J5cAOWgEYtUZZ3ofOp3k2Nvb21xSFKc4KfAXCqjNfDeojlhsc/Jfl+tImiQQk0ebotjlu/7/Qsl4pxghV5GzBrm6uE/y2F9rvFiWGK0L9aB/mlDM1OVwRZKGEwrVNdUhV+exRuF0tdiKBA24LYlpSNrkBpqf+STelsRNACOLWcC2p+84DcUAwtJQzBYGWcQco+9XdOpm19RxyGk5UHLvaOoSpsCNCBV1YxazZ8TiY0MnN2HLjoYbpUZqvAZh52zKxtUAHdSTdaLszlYUD5OP7g676JDD6cWH0kNVAHEnNqIhCDZGXI2Cp80RV8MH6HLleBNYO+INRON7Q6lL72FwamjTta7xAj8p2KJHq4E8CQUtjbrUVu1TGu0NbP4cbwSW9frjdmG0Vw2r9rmFxRYNl9AjXLD1VhR29e0XcNqgXq22fs2ItpzeklO92do4HLlCyDl7I95RcNdt0UGncrboIEC0iPLP/bgY8G4L32ZJOjlv2CRImPBb3xuDifzvA5+eoyDXhI4S4gAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Question 11.Find the probability of getting 5 exactly twice in 7 throws of a die.
Answer:In this question, let us assume X represent the number of times of getting 5 in 7 throws of the die
Also, the repeated tossing of a die are the Bernoulli trials
Thus, probability of getting 5 in a single throw, ![](data:image/png;base64,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)
And, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Clearly, we have X has the binomial distribution where n = 7 and ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADEAAAAgCAMAAACMyGAQAAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6Ojo6OmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZrbbZrb/kDoAkJBmkNv/tmYAtmY6tv//25A625Bm27Zm27aQ29u229vb2////7Zm/9uQ/9u2//+2///bgGTBTAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAz0lEQVQ4T91U0Q6CMAzsAJkKioDIcCAC0v//RDs0EaOJrU/Ge+CB9dq7WwHgx4BVKFNUr1MhA8D+BwPNohOFZRVhK6J8W4yVCqHR3p7doC57/5CD8Y9sCljnxnrlg2GcxQnzt7PzgCJ7Yryfdm8CgIULTKJq0GR6TOY79kFV77RODy6syuGkJVdpolR5Obc/1Y0J/+5ubYelwEGTRZ3EM1Ybt+xGxdydF3/lWETZilTxMej43BaSv8m4o4xswB9BCxh3KJoBl1TxYxJIeS29AhDrCrm0u9/mAAAAAElFTkSuQmCC)
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, probability of getting 5 exactly twice in a die = P (X = 2)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 12.Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Answer:In this question, let us assume X represent the number of times of getting sixes in 6 throws of a die
Also, the repeated tossing of die selection are the Bernoulli trials
Thus, probability of getting six in a single throw of die, ![](data:image/png;base64,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)
Clearly, we have X has the binomial distribution where n = 6 and ![](data:image/png;base64,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)
And, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, probability of throwing at most 2 sixes = P (X ≤2)
= P (X = 0) + p (X = 1) + P (X = 2)
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
Question 13.It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective?
Answer:In this question, let us assume X represent the number of times selecting defected articles in a random sample space of given 12 articles
Also, the repeated articles in a random sample space are the Bernoulli trials
Clearly, we have X has the binomial distribution where n = 12 and ![](data:image/png;base64,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)
And, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, probability of selecting 9 defective articles ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIwAAAAmCAMAAAAhmYOEAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOmZmOmaQOma2OpDbZgAAZgA6ZgBmZjoAZpDbZrbbZrb/kDoAkDo6kGY6kGZmkLbbkNv/tmYAtmY6tpA6trbbttv/tv//25A625Bm27Zm27aQ27a229u22////7Zm/9uQ/9u2//+2///bjLhdnwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC3klEQVRYR+2Ya3fTMAyG5bGxcBkjJStdYRlsaYEmsf//v8NOfJF8i096tg8c8qlNH0tvJEtWCvD/ev0IHN6zi7tCt0NTzhaaJJi4/w7Hi4eipbzewlAVoeuh4V2ZmP7NM0C73k/JynG/LcEAlOhxV8amqfHbk/djf3syd1p2icSEKDi2Y+ymOVPM8CFMQ/dRhlxfv6qv5mMMBczyzXlieG184WrorqbY8M2D3JQGcCiIx2vrVrPy+9h8Pk9Ma6zO1aAdi91s9lChcrUoHG4aJ8awYoczukaVq5W5GsyjTd/oRcqqc2Igwq6RIr1bo3M1mK/mcZFVh8qbWEyEDcTovP5pGLtNKXUbQtpX1WCT3vmhwSgVAwEbuNN55ZtcE/XMqA2rL+pbxYKow5FBezydHrsg3UTbuWr0hauB115lEFS0b9HCgI1osmI6bXfcV4w0MV6jfUirQeyITiCozChjTqzPxuJjxBz1lhmqqyc4MtvDZAuv0o2hpZsmh4LHZsR0Wguv1bPyL6hme6zMM9HR0zqHgsdGxOi89lJLr6LRhrPAq4nReeW1yq8UQ5KuteeeqKNRyz68x0KrfE5XYhiKRSHnweOzYrI5jO2g2CYzHuyDzB+m5QkxMdRnFw+DaPm91J5ZShMS42aFlxKzGJo5TfJYRrNC0PORlaDPoP7kOyvoM3RJz+5A/JC7G80K2aZHO3C+6VF2MTDToMQuP5305DydA8KODMF6/6cMmjOzqAvNCvSgxCuDSKTR7KmyqGYebqf6NU1JDz9ib8ZM+4uxlkadlQLPIWJnBXP4m6G2uz7peSMIRBqFTNAW5eFZwY720xGv5v95iERvAsZeCo2xixqigHU6i1HZm2Zd9/5hl6XQGLtOjLVExUQC4wR6uqPsSjV66CdpSsz7MVSndaVzf5m4V2/WeviRW1FuYHkrajyCQopdq+73s3pNmYZaLv/w2cL4M2UqQDPsWjn/6Lq/cONKOalPg10AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 14.In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
A. 10–1
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:In this question, let us assume X represent the number of times selecting defected bulbs in a random sample of given 5 bulbs
Also, the repeated selection of defective bulbs from a box are the Bernoulli trials
Clearly, we have X has the binomial distribution where n = 5 and ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADgAAAAgCAMAAABw3UvaAAAAAXNSR0IArs4c6QAAAGlQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OmZmOmaQOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkGY6kJBmkNv/tmYAtmY6tv//25A625Bm27Zm27aQ2////7Zm/9uQ//+2///bMRspygAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA00lEQVRIS91UXQ+CMAzshgJ+ICJKcajA/v+PtBugGHmgNTHGeyCE7Nre0R3Az8OWoWjGapXKiADm/4kWlzeJrUYRthLm5xxbqhAukT5wS1VFHZwywODMZYJxYo0unkR0DniMv74VxgX5+0Kc7v2o5l8AbO7cFYzaRORLm4xXd96otdPhH0wYlcE1EiwDrlOlM2Y3Ot4m7F/fNWni+er6vLFHpfccW4a8MeGtiQHVZv798nnjxNmcZ0pH3JE4/CqRP2qfN3QryBwGhrxpZRvD6DR59A7htQtA6QcJPgAAAABJRU5ErkJggg==)
And, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, probability that none bulb is defective = P (X = 0)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEgAAAAvCAMAAAB3wnyXAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOjqQOmaQOma2OpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kGZmkLaQkLbbkNv/tmYAtmY6tpA6ttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bjKMBoAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAByElEQVRIS+1W23KDIBCFpBfbJjW92LSJaYqVyP9/YRdZLioKzDCdPtSHTAbXw9mz7FkI+X9mFWCU0tUxg0DsNQOIhMgHBKltk0g1O0qvz75PeJGSHlsdiKjXXx4kUT2GKYl3RaIrZbD6hYfrbMTbkUQxOukve/qiuvlWSHpdnKD6L2FC7BZjFBcLJKoUXQi/M0etXh+J+KSaEWm9cs1RqzUhILOHon2UZiFKYY3rEEKNC1ufFErMJILQzOkqU8Gw0pPQzmYGX9fjbWYR21FXN8UWi99/Mn49z4wNCsOL+8MgljuCLaZnD40/TFS2potAQ0U8obEiBan7W3i6Y0sDXeAehqXcsgEFNwzu5DnGXuZToBp8Vz3uEczOyOzi/FFHO1SNP5takHlQREwtG1C2FnGbtnlSU0A694Met7FNa23ksqPKmuTSpUKbirYRoo2Nb874tzdFrV0wddMOjpciUG8cOuV4q3XMXwGpZLtSuX6sRBBqx5ELhNoljCNnYnuAUgYkMSN7mJo0/aSRbSkNxZZzO4kQqIQTB4GYNI6+/InXGnOhwv6UBVNa64vW4kBzX8qrn9gX4GlXz7AOZ5xuAMdc/aKBfj3wB2aAJUN1fZkqAAAAAElFTkSuQmCC)
∴ option C is correct
Question 15.The probability that a student is not a swimmer is 1/5. Then the probability that out of five students, four are swimmers is
A. ![](data:image/png;base64,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)
B.![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. None of these
Answer:In this question, let us assume X represent the number of students out of 5 who are swimmers
Also, the repeated selection of students who are swimmers are the Bernoulli trials
Thus, probability of students who are not swimmers ![](data:image/png;base64,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)
Clearly, we have X has the binomial distribution where n = 5
And, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, probability that four students are swimmers = P (X = 4)
![](data:image/png;base64,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)
∴ option A is correct
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes? (ii) at least 5 successes?
(iii) at most 5 successes?
Answer:
We know that the repeated tosses of a dice are known as Bernouli trials.
Let the number of successes of getting an odd number in an experiment of 6 trials be x.
Probability of getting an odd number in a single throw of a dice(p)
Thus,
Now, here x has a binomial distribution.
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
= 6Cx (1/2)6-x(1/2)x
= 6Cx (1/2)6
(i) Probability of getting 5 successes = P(X = 5)
= 6C5(1/2)6
(ii) Probability of getting at least 5 successes = P(X ≥ 5)
= P(X = 5) + P(X = 6)
= 6C5(1/2)6 + 6C5 (1/2)6
(iii) Probability of getting at most 5 successes = P(X ≤ 5)
We can also write it as: 1 – P(X>5)
= 1 – P(X = 6)
= 1 – 6C6 (1/2)6
Question 2.
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.
Answer:
We know that the repeated tosses of a pair of dice are known as Bernouli trials.
Let the number of times of getting doublets in an experiment of throwing two dice simultaneously four times be x.
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
= 4Cx (5/6)4-x(1/6)x
= 4Cx (54-x/66)
Hence, Probability of getting 2 successes = P(X = 2)
= 4C2(54-2/64)
Question 3.
There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item?
Answer:
Let there be x number of defective items in a sample of ten items drawn successively.
Now, as we can see that the drawing of the items is done with replacement. Thus, the trials are Bernoulli trials.
Now, probability of getting a defective item,
∴ we can say that x has a binomial distribution, where
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
Probability of getting not more than one defective item = P(X ≤1)
= P(X = 0) + P(X = 1)
= 10C0 (19/20)10(1/20)0 +10C1 (19/20)9(1/20)1
Question 4.
Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is a spade?
Answer:
Let the number of spade cards among the five drawn cards be x.
As we can observe that the drawing of cards is with replacement, thus, the trials will be Bernoulli trials.
Now, we know that in a deck of 52 cards there are total 13 spade cards.
Thus, Probability of drawing a spade from a deck of 52 cards
Thus, x has a binomial distribution with
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
(i) Probability of drawing all five cards as spades = P(X = 5)
(ii) Probability of drawing three out five cards as spades = P(X = 3)
(iii) Probability of drawing all five cards as non-spades = P(X = 0)
Question 5.
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one will fuse after 150 days of use.
Answer:
Let us assume that the number of bulbs that will fuse after 150 days of use in an experiment of 5 trials be x.
As we can see that the trial is made with replacement, thus, the trials will be Bernoulli trials.
It is already mentioned in the question that, p = 0.05
Thus, q = 1 – p = 1 – 0.05 = 0.95
Here, we can clearly observe that x has a binomial representation with n = 5 and p = 0.05
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
= 5Cx(0.95)5-x(0.05)x
(i) Probability of no such bulb in a random drawing of 5 bulbs = P(X = 0)
= 5C0(0.95)5-0(0.05)0
= 1× 0.955
= (0.95)5
(ii) Probability of not more than one such bulb in a random drawing of 5 bulbs = P(X≤ 1)
= P(X = 0) + P(X = 1)
= 5C0(0.95)5-0(0.05)0+ 5C1(0.95)5-1(0.05)1
= 1× 0.955 + 5 × (0.95)4 × 0.05
= (0.95)4 (0.95 +0.25)
= (0.95)4 × 1.2
(iii) Probability of more than one such bulb in a random drawing of 5 bulbs = P(X>1)
= 1 – P(X ≤ 1)
= 1 – [(0.95)4 × 1.2]
(iv) Probability of at least one such bulb in a random drawing of 5 bulbs = P(X ≥ 1)
= 1 – P(X < 1)
= 1 – P(X = 0)
= 1 –(0.95)5
Question 6.
A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
Answer:
Let us assume that number of balls with digit marked as zero among the experiment of 4 balls drawn simultaneously be x.
As we can see that the balls are drawn with replacement, thus, the trial is a Bernoulli trial.
Probability of a ball drawn from the bag to be marked as digit
It can be clearly observed that X has a binomial distribution with
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
Probability of no ball marked with zero among the 4 balls = P(X = 0)
Question 7.
In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers 'true'; if it falls tails, he answers 'false'. Find the probability that he answers at least 12 questions correctly.
Answer:
Let us assume that the number of correctly answered questions out of twenty questions be x.
Since, ‘head’ on the coin shows the true answer and the ‘tail’ on the coin shows the false answers. Thus, the repeated tosses or the correctly answered questions are Bernoulli trails.
Here, it can be clearly observed that x has binomal distribution, where
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
Probability of at least 12 questions answered correctly = p(X ≥ 12)
= P(X = 12) +P(X = 13)+…+P(X = 20)
Question 8.
Suppose X has a binomial distribution . Show that X = 3 is the most likely outcome.
(Hint: P(X = 3) is the maximum among all P(xi), xi = 0,1,2,3,4,5,6)
Answer:
As per the question,
X is any random variable whose binomial distribution is
Thus, P(X = x) = nCxqn-xpx , where x = 0, 1, 2, …n
It can be clearly observed that P(X = x) will be maximum if 6cx will bw maximum.
∴6cx = 6c6 = 1
6c1 = 6c5 = 6
6c2 = 6c4 = 15
6c3 = 20
Hence we can clearly see that 6c3 is maximum.
∴for x = 3, P(X = x) is maximum.
Hence, proved that the most likely outcome is x = 3.
Question 9.
On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing ?
Answer:
In this question, we have the repeated correct answer guessing form the given multiple choice questions are Bernouli trials
Let us now assume, X represents the number of correct answers by guessing in the multiple choice set
Now, probability of getting a correct answer,
Thus,
Clearly, we have X is a binomial distribution where n = 5 and
∴
Hence, probability of guessing more than 4 correct answer
= P (X = 4) + P (X = 5)
Question 10.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is 1/100. What is the probability that he will win a prize
(a) At least once
(b) Exactly once
(c) At least twice?
Answer:
(a) In this question, let X represents the number of prizes winning in 50 lotteries and the trials are Bernoulli trials
Here clearly, we have X is a binomial distribution where n = 50 and
Thus,
∴
Hence, probability of winning in lottery atleast once
= 1 – P (X <1)
= 1 – P (X = 0)
(b) Probability of winning in lottery exactly once
(c) Probability of winning in lottery atleast twice
= 1 – P (X < 2)
=
Question 11.
Find the probability of getting 5 exactly twice in 7 throws of a die.
Answer:
In this question, let us assume X represent the number of times of getting 5 in 7 throws of the die
Also, the repeated tossing of a die are the Bernoulli trials
Thus, probability of getting 5 in a single throw,
And,
Clearly, we have X has the binomial distribution where n = 7 and
∴
Hence, probability of getting 5 exactly twice in a die = P (X = 2)
Question 12.
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Answer:
In this question, let us assume X represent the number of times of getting sixes in 6 throws of a die
Also, the repeated tossing of die selection are the Bernoulli trials
Thus, probability of getting six in a single throw of die,
Clearly, we have X has the binomial distribution where n = 6 and
And,
∴
Hence, probability of throwing at most 2 sixes = P (X ≤2)
= P (X = 0) + p (X = 1) + P (X = 2)
Question 13.
It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective?
Answer:
In this question, let us assume X represent the number of times selecting defected articles in a random sample space of given 12 articles
Also, the repeated articles in a random sample space are the Bernoulli trials
Clearly, we have X has the binomial distribution where n = 12 and
And,
∴
Hence, probability of selecting 9 defective articles
Question 14.
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
A. 10–1
B.
C.
D.
Answer:
In this question, let us assume X represent the number of times selecting defected bulbs in a random sample of given 5 bulbs
Also, the repeated selection of defective bulbs from a box are the Bernoulli trials
Clearly, we have X has the binomial distribution where n = 5 and
And,
∴
Hence, probability that none bulb is defective = P (X = 0)
∴ option C is correct
Question 15.
The probability that a student is not a swimmer is 1/5. Then the probability that out of five students, four are swimmers is
A.
B.
C.
D. None of these
Answer:
In this question, let us assume X represent the number of students out of 5 who are swimmers
Also, the repeated selection of students who are swimmers are the Bernoulli trials
Thus, probability of students who are not swimmers
Clearly, we have X has the binomial distribution where n = 5
And,
∴
Hence, probability that four students are swimmers = P (X = 4)
∴ option A is correct
Miscellaneous Exercise
Question 1.A and B are two events such that P (A) ≠ 0. Find P(B|A), if:
(i) A is a subset of B
(ii) A ∩ B = φ
Answer:It is given in the question that,
A and B are two events such that ![](data:image/png;base64,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)
We have, ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
Hence, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 1
(ii) We have,
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
= 0
Question 2.A couple has two children,
(i) Find the probability that both children are males, if it is known that at least one of the children is male.
(ii) Find the probability that both children are females, if it is known that the elder child is a female.
Answer:(i) According to the situation, if the couple has two children then the sample space is:
S = {(b, b), (b, g), (g, b), (g, g)}
Let us assume A denote the event of both children having male and B denote the event of having at least one of the male children
Thus, we have:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(ii) Let us now assume C denote the event having both children females and D denote the event of having elder child is female
∴ C = {(g, g)}
![](data:image/png;base64,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)
And, D = {(g, b), (g, g)}
![](data:image/png;base64,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)
Hence, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 3.Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male?
Assume that there are equal number of males and females.
Answer:It is given in the question that,
5% of men and 0.25% of women have grey hair
∴ Total % of people having grey hair = 5 + 0.25
= 5.25 %
Hence, Probability of having a selected person male having grey hair, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADwAAAAgCAMAAAB5NuugAAAAAXNSR0IArs4c6QAAAGNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6tv//25A627Zm29u22////7Zm/9uQ//+2///bOS4+mAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA60lEQVRIS9WUyxaCMAxEG98P1KKAFi3t/3+lCUfAclCbdGVWLJhmOkmvUv9RBgBmpdCr0UIhydLEaHuX0LzZJFj3+V7W2p9LNeps8SIAi+L3gf6KkzqF/xkcnb8I52fmN6UsiILw+eohFruM8quoPb/IMEbRT6BqA6SKiIG2HZZfwu4PG3+QYbqysFy2DpUc20nLWskG3LrFuCJCFUaSLHtD1/3Y4oABs4E+7lCoGq/J4FH4a7PliQN0GVxksh0Ps2Eb6peIsR89ukzXMBJmHbosvlqLWqsnYPZppB26UOwyell6CmbJC8E54AmQqQwgAk6OAwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Question 4.Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?
Answer:It is given in the question that,
90% of the people are right handed
Let p denotes the probability of people that are right handed and q denotes the probability of people that are left handed
∴ ![](data:image/png;base64,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)
And, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACYAAAAqCAMAAADoIdnnAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOjqQOmaQOpDbZgAAZjoAZjo6ZpDbZrb/kDoAkGY6kNv/tmYAttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bx4OD6QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAArklEQVQ4T+2T3Q6CMAyFV0H5UQQEhoh7/9dkTQcuxtKFEG+0FwsXH+3O6Y5S36q+uMmjxgtEnYg9slYHYLbPH3szUx/kLZg6AYD4Ku7hV4HG2kMV4KZn0vKb9xHs4dahwQP2BufAmxogbZnuS+BNdbyPFZPqV+AHdHyAkrusC3yDjZ7n0zqGMxHD82NRN8Lo3AsT76achFxQqtGLNUPoiaJIVoEXeLsPyDidm1/IBPskCs6fiaJVAAAAAElFTkSuQmCC)
Now, by using the binomial distribution probability of having more than 6 right handed people can be given as:
![](data:image/png;base64,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)
Hence, the probability of having more than 6 right handed people:
= 1 – P (More than 6 people are right handed)
= ![](data:image/png;base64,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)
Question 5.An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15 bear a mark 'Y'. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that:
(i) All will bear 'X' mark.
(ii) Not more than 2 will bear 'Y' mark.
(iii) At least one ball will bear 'Y' mark.
(iv) The number of balls with 'X' mark and 'Y' mark will be equal.
Answer:(i) It is given in the question that,
Total number of balls in the urn = 25
Number of balls bearing mark ‘X’ = 10
Number of balls bearing mark ‘Y’ = 15
Let p denotes the probability of balls bearing mark ‘X’ and q denotes the probability of balls bearing mark ‘Y’
∴ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE8AAAAgCAMAAABkSNU7AAAAAXNSR0IArs4c6QAAAHJQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OmZmOmaQOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kGY6kJBmkLbbkNv/tmYAtmY6tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ//+2///biiBbiwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABSUlEQVRIS91VbVPDMAhOutnqZlc7dTYz61yX/P+/aKAv66YQ7ox+kA+93gEPPBBAqX8gfp8DC/+qsyeSzkel9VpE9vBQIZ7NT+f7hnBxmzfVZpT2xskCniu3yj8/MinQ0b7F24ToBjMlxHLB5j59fjG8VlY+rJyArxXDeXN3CoBmyfRDdQGu20oabHWQUBpX6aymHFwJViI8Scxfs/F7nau2yFJlemi6xa5WZvGeKmULtbbjqBgoKwo3PJxVeAYzvK9ZTgH6H4LGYBW2Bo5kOr7nIrTCleNc/phvB3XCTxqxulbHQroZ4jHNihujuP+NBWzJlCLYqeNlgL1A13nQxjsxXQbLMhm0Rq9ht0UEWYjwYki9Hi8DMKJ3MK+9DjNdBnz8pPDai9vlMvD3ktdOeMNl8C+NojPgtXNK42UIm5e5H7xW1qU/sPoEMJsTuNSFsxEAAAAASUVORK5CYII=)
And, ![](data:image/png;base64,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)
Now, 6 balls are drawn with replacement. Hence, the number of trials are Bernoulli triangle.
Let us assume, Z be the random variable that represents the number of balls bearing ‘Y’ mark in the trials
∴ Z has a binomial distribution where n = 6 and ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, P (All balls will bear mark ‘X’) = P (Z = 0)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFMAAAAvCAMAAACojAaIAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjpmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kGZmkNv/tmYAtmY6tpA6ttv/tv//25A625Bm27Zm27aQ29u229vb2////7Zm/9uQ/9u2//+2///bI9EGrgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACFklEQVRYR+VX2VbDIBCFujRqo9FuWE2tadWGhP//PYEBskEW6JPOS0+X3Lmz3Zki9C/ttFqcLxs42z9eGBGhbH5ZjhyNkcXqPih29rXE+Oa9xoxG8c83CeGa4g0qkqvPCrR85h6yW8+UsrcDSgWhDG8RjRUsI/GZefPcPymYnGMi867g2Yg9aVb1TUXsjHDgQKN3ujI0kmh5Pat+4DKTwhiBHOhXPzjxlKFZ1SOYqOmWdK7rUSa6aH5UzfMSOwew1LcrgUM+gwrJFLAUMPWHdpqDapWpIqdYGmDSyB38sFoxYguzb3yG1apMrCLhTmhXrYpdhPH1psqTI0o5UFbrqBWNbg/oJIZam5zwrmWqct1v2mpVJiJ55UuNw2TMtlqlXe8OQg5XgnZTrWz1GIWp+oy3WoeUzfkozJ4JtRVTY0LHC4NBspeuA25t71E83bHXMNkOz9bS6SjMwdiFTGTzs5LNyb3Uws/xGrEPXroy2WoxD8VERzGY/LKQswArozWbR7Wa3LPpykMN0/Qs5f6q/psuyrXYTeHUzgQaPqcCp6FXm9bkBmafJjuDX+KZEjyzO2C3g/Xvjp7Wgq/0jmvwnJ7Oph+1i2n0EOlbMXQX65uBvfJbkYDuBN8MqlEleahN+G1THTciuWIQwmmq/wLyWpA8L3ErwhUr9kpBxLo0N+1gz/T9QNzexYqP5uLAuerbOwjyjzz8C8t8L4EMa8/LAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
(ii) Probability (Not more than 2 will bear ‘Y’ mark) ![](data:image/png;base64,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)
= P (Z = 0) + P (Z = 1) + P (Z = 2)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUsAAAAXCAMAAAC1Up2UAAAAAXNSR0IArs4c6QAAAKtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZpCQZpDbZrbbZrb/kDoAkDo6kGY6kGaQkJBmkLaQkLbbkNv/tmYAtmY6tpA6tpBmtrZmttv/tv//25A625Bm27Zm27aQ29uQ29u229vb2////7Zm/9uQ/9u2//+2///b6YQ2UgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAETklEQVRYR+1Yb18TMQzuDUFOwaEgKGPiHw4Vjinuxt33/2SmbdKm16Zl589X0DewLcmTPE3Spko9r2cGnhl4cgyszubrKUGX9IbLanaeNFzSjJTaqqpmVzknh+t94eetwZTa1Bc5rPvTqjpKCQzXbwtMDj8Ob71mO78xH5heWkC1++vNqwQBOUTJlI8tkCBn1N2b0zSX2fDSttTw6TXiJd3pT76pVXJnW2k/ib5hGZK9OTBAXk8Q6I8v1LB8F+9fBlEwpVrH5UgCnQn8CRBz4Um22sXS4knuQOYmk2Q5PzugGn+4rKvqxSJwJqbD2BmcniCg+hMQa+Kd8ppKjQAlU0rXuKmqSMIFlSYtAyba6o4Gy6XoDmRSIkegNRz9uV/aiDf13o1aVWGv6HZYgVuW27010xMEQi7RO4viEMeAkimrpv2KJbQz4zrxEWTAJFv9h9thufgMVmV3VmZjx8tkj/WnP9Z/+vcBdwPyzPV0Ljg9SQDMsRpnXHrEMaBoyiVJQoISk/ccz6UMBlkX1YyxpWugqoAK2Z02SSUoHK1RqUn1UzzShutqX63qmct9p+fPvFVd7X491Ltii6PZ82cP49IjjgElLDgMrmxeOrAxlhqal3SEMiwWXhSdtRUFZvdN/yRG1gGVXeqkf9AHvE3L1CnUWYLvrrqdLwvV2ILXXZD0UAD2c7ZQHTBOAqo/reAru3h8pBkBilgQsDVFYBGWSSbsYRzLuRlHZ23FgcGXv2sdpxRZf6zBgMvGZLBe4yzsRo3SkuANaldbqwQZ52qeBDY1/E5byQUiLkk1AixikTfbY8EujxPJ7YsQmAs9j+aI4P9gzo1+a+noMfQkuCQBo07+BVxK2xcBFrEUSmyPRRXFonOOC4ERmFVNRpbk0VQhSzYvRDawBWKNx3lpe27gH0cK6o7KfgxYxMKItsdKRYdoYmBEn4wm1jjjkg9/voShRqjp8FsjCvTHUOLORnStzHOJA2ARC8PbHotzSRMgopm2lAqMuCygZWq80V2RDX/UAQ0y541soIBBbLH9mw/BSnCJZQeANAAWsbAfT8DyYG4CRDQxMGr+BbQkl111robvQNnoYmhP5rZaqF/6eIEVDFB4oYI/69VHuFJGApmzBwFpIKULnYiFV74JWHD2ODAMANFKYBozF1m6Z97pARLGyXD4w8t/M/c3HBo1rBkU6OrZubs3R603lZeKAN1wX8RCsAlYDIwmQIsmB/a4yKTzx34fcml7s85VV9XhXM+mViQ5OfhnMXFoKWKxEXkyllI4ARq0TGB8Hp+KNnrgMY8ljB7/NIP0+NcUe+REAvmtMx0Ex4Qiln+6mYyl3ASo0XKBsXeiyWh8+INA9SOeu0Yr92ToGaJXPnvkJARKZPoBsIhlvDGFMhFLsQkQbP10I0vK73+OLBj+kIbGzpn5tal37TvxlosPgLqD/U8s2AScAK2TjwKDI31aZFsS8WTF/wJM0M3SHIYYvgAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
(iii) Now, Probability (At least one ball will bear ‘Y’ mark) = P (Z ≥ 1)
= 1 – P (Z = 0)
= ![](data:image/png;base64,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)
(iv) Probability (Having equal number of balls with ‘X’ mark and ‘Y’ mark) = P (Z = 3)
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Question 6.In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is 5/6. What is the probability that he will knock down fewer than 2 hurdles?
Answer:In this question,
Let us assume p be the probability of player that will clear the hurdle while q be the probability of player that will knock down the hurdle
∴ ![](data:image/png;base64,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)
And, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Let us also assume X be the random variable that represents the number of times the player will knock down the hurdle
∴ By binomial distribution, ![](data:image/png;base64,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)
Hence, probability (players knocking down less than 2 hurdles) = P (X < 2)
= P (X = 0) + P (X = 1)
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Question 7.A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.
Answer:From the condition given in the question, it is clear that:
Probability of getting a six in a throw of die = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAADqQAGaQAGa2OgAAOjo6OmZmOmaQOma2OpC2OpDbZgA6ZjoAZjo6kDoAkNv/tmYAtmY625A625Bm27Zm29u229vb2////9uQ/9u2//+2///bmbpMdwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYklEQVQYV2NgwAAyQiwgMRF2XjDNwCCMn5YRZJYAqWIEAm5M43CIgFQDAdHqkRSK8nEALZQR4gJZC3OdDD8HHxtQXJKVU1yMn4VBmkcAKMfEIMPPKSED5DNI8TIycoJ1IAMAZUYDtd9yyncAAAAASUVORK5CYII=)
And, probability of not getting a six = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6ADqQAGa2OgAAOjo6OmaQOma2OpC2OpDbZgA6ZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmY625A627Zm29u229vb2////9uQ/9u2//+2///beWS0qAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAa0lEQVQYV41P2Q6AIAwbKoIHHoAHyv9/p0WMGo2Jfdiybt06oicsYyzRRLaNnTODl5FyPPZ8V5Lv9V57A1nz2vZJYFvAf8E1OSgx47gpEOEvO7wIlYN3XE5jl9FahR9SeJSzR01LDf+74o4NwRsEMLBHs7EAAAAASUVORK5CYII=)
Let us assume, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHgAAAAgCAMAAADuZ0x6AAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6Ojo6OjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6Zma2ZpCQZpDbZrbbZrb/kDoAkDo6kGY6kGaQkJBmkLaQkNv/tmYAtmY6ttv/tv/btv//25A625Bm27Zm27aQ29u229vb2////7Zm/9uQ/9u2//+2///bPnG7NAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB5klEQVRYR+1WaVPCMBBNOKQeIBVUrBdFDVXb2vz/P+fuJj2oNNnOdMYZx3wiy9u319uAEP/nSAf0bjZgX5SUcrTlEO4vVoMGvuMENRj1NwJDqxfMogetGGLmAbPdQwfW0ZJVso5PUjewCCVXBvp+y60Y9S89KRYhrwYh9A62acMqmANiT41D1geTjd/6wAELlc9EEoyYMksCOXk+T8FrmpZeaHuYViqwEF8e+202ftqImJexgulkkKnx0hGGK202UuvqiK9QOKp8PWNUEp0j72keANSMk7xikHLDhkFaV+Hgi03anc+2zQNZqS/5KULxI60j2bLKvXX9WXBFZ5aZ1Wod4bIqjEReWHvDhkFaV9egqXNFWO6/q9W0rIaavFDKDRtGsddaat181KW6Va4ciZWGaxxwSGTbV+9LA+Jiwu+U3Ih3FI3/QLFpchMu82vT73imH19rm201QjjLGc9X7CcsC0a3oNNFakcpJy+ispnE8Wrk5zkFKzsfy+H3qh5xtyMru2Q99/w8HQTA9XYdomPISu8u+4Q1eus+lg4n5mlk378BeQCj7z5sOh3N12e9Wu0shE+XB4vPD3o1Bjl8uuIKloMlVFZilo6B1VG5twywH9KD7msFf4R9CvRHLBED0/ED/zLyGxjgL/LCIguyAAAAAElFTkSuQmCC)
Now, we have:
Probability that the 2 sixes come in the first five throws of the die ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADAAAAAlCAMAAAAzx5qdAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjo6OjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjqQZmaQZma2ZpDbZra2ZrbbZrb/kDoAkDo6kGY6kLb/kNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ29u229vb2////9uQ/9u2//+2///b14tOLwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABhElEQVRIS81U61qDMAxt2RwqgpeBV1AHIjql0vd/OXNSyoANUPdj5vvGCsnpSdKTCvGfTD9JORtNaH2TG//bbUnPaplX4RiguGi8ygNCiLVHD50tsCZC566N56BCSums6B/g6lL6RHcWMqBYlOqYXFkM13VpYgq8kamTV3zUiQlldCx0QjFVFNNvRS+ItQD9uBJVWBdtAFe0e8qr6J4wFkApBUiY4bX1AEK5SKYJUW4D7wBsSl0GRhIeKVnT6RF6ls5N0cgfdXDRiGMGFG33pywlig2l80DfXrAVuoQYnVFX8bV1Jpta+qsC1RqzBzcczJ6eNCaiD+dGj35jh8t0L2b1PAJvzqqeY5FSR1rK3lIDiaorh2yj0h08tWTeIx/6ZcERw2wYwyE6OzdXRC1pzXO40+oxtn6e42U+BdCJH51ySsBXiZReMzt9GmZQbvD5kYClM/jDKfH1UcyJoj3HA0XwGCcB3VdgaM3xUNUc8kX3VdC0dUIjrTH+4RxvSWMvFf4J/A3K/yl1ZFG6nAAAAABJRU5ErkJggg==)
Also, Probability that the six come in the sixth throw ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 8.If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays?
Answer:We know that, in a leap year there are total 366 days, 52 weeks and 2 days
Now, in 52 weeks there are total 52 Tuesdays
∴ Probability that the leap year will contain 53 Tuesdays is equal to the probability of remaining 2 days will be Tuesdays
Thus, the remaining two days can be:
(Monday and Tuesday), (Tuesday and Wednesday), (Wednesday and Thursday), (Thursday and Friday), (Friday and Saturday), (Saturday and Sunday) and (Sunday and Monday)
∴ Total Number of cases = 7
Cases in which Tuesday can come = 2
Hence, probability (leap year having 53 Tuesdays) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAAgCAMAAAA7dZg3AAAAAXNSR0IArs4c6QAAAEtQTFRFAAAAAAAAAAA6AGa2OgA6Oma2OpDbZgAAZgA6ZjoAZrb/kDoAkDo6kDqQkLbbkNv/tmYA25A627Zm29u22////7Zm/9uQ//+2///bZDM6kAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAc0lEQVQ4T7VSSRKAIAwr7ruCIvz/pVJHT6Y4zmCuaZI2QJQKa6tUCc1cM5HJZiloL0RKV5LI4KgwrkXGBsZ2yNHVKgBSqeoBPguHnpDLuWXXYHybL4Y/3vWw9j2vn6NIP27kB/ED8JNhxEQwiW1Si95LPAAIWAMaljnS+gAAAABJRU5ErkJggg==)
Question 9.An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.
Answer:It is given in the question that,
Probability of failure = x
And, probability of success = 2x
∴ x + 2x = 1
3x = 1
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
Let us now assume ![](data:image/png;base64,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)
Also, X be the random variable that represents the number of trials
Hence, by binomial distribution we have:
![](data:image/png;base64,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)
∴ Probability of having at least 4 successes ![](data:image/png;base64,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)
= P (X = 4) + P (X = 5) + P (X = 6)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 10.How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%?
Answer:Let us assume that, man tosses the coin n times. Thus, n tosses are the Bernoulli trials
∴ Probability of getting head at the toss of the coin ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAAgCAMAAAA7dZg3AAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAb0lEQVQ4T72RSw6AIAxEi39FhYoi3P+ignHZCibE2b4y0w4AxeR1w3htneQQAP6HvKoPekUUQUOxKvKMVAy9Va2pF8/g+9gXw1RgUb5LIXrS0U0LGP78s2WbQfa7DB0V8pElNhA7Uyu6MXZPooyOLjHfArzqNRIZAAAAAElFTkSuQmCC)
Let us assume, ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
It is given in the question that,
Probability of getting at least one head > ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAAgCAMAAAAhSXi5AAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OmZmOpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkGY6kNv/tmYAtmY6tv//25A625Bm2////7Zm/9uQ/9u2//+2///bvKnIkgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAoklEQVQoU8WR3Q6CMAxGO/+mDCegU8Sxvf9b0nYrRNQYvaE3X3LStUsPwO/lrVqdAOKZQyqUFfh9Da3u/OEy4sf6BuBMKGuIjRkxtfSNDkfsdHqa0ipVWPOCsQHZfAjS3uJMt31aiYs2Fb3Af1IuU+ptLfOXD1vjlS8vInPeC8tYRHImQEfNIlNOOItM5r7i+ZDodh0+EpGcuJBOiH5F5B9CB7aVDK64soP+AAAAAElFTkSuQmCC)
∴ ![](data:image/png;base64,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)
1 – P (x = 0) > 0.9
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(i)
Hence, the minimum value of n satisfying the given inequality = 4
∴ The man have to toss the coin 4 or more times
Question 11.In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins / loses.
Answer:For the situation given in the equation, we have:
Probability of getting a six in a throw of a die ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAAgCAMAAAA7dZg3AAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADqQAGaQAGa2OgAAOjo6OmZmOmaQOma2OpC2OpDbZgA6ZjoAZjo6kDoAkNv/tmYAtmY625A625Bm27Zm29u229vb2////9uQ/9u2//+2///byGMlQAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAdklEQVQ4T71S2Q6AIAwb3uKFgrf7/98UjY8baELsa9d26QYQDKhTxmssWo4CMP9ROCQrvaIRFnWwKt4ZDVfojaj3KZ5B99gXQ19gYH7qSrp61BVzE8dnoCq7nDbcMrnMivypo7E1m5iKQyVXpFWwt0JIbkdfTSeMugRn41GnqAAAAABJRU5ErkJggg==)
Also, probability of not getting a 6 ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAAgCAMAAAA7dZg3AAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADqQAGa2OgAAOjo6OmaQOma2OpC2OpDbZgA6ZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmYAtmY625A627Zm29u229vb2////9uQ/9u2//+2///b9FykOwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAhUlEQVQ4T62SyQ6AMAhEqVqte13qUuX/f1MSPXgAl6RcYd7ABIBQ5ZRSUc/SXCuaPLUIaESh1yIUbcHJsOtBUOFIuzehkvjKGejAs4TcbqBr8Bn9B/h1yUBzU52tLArHnG8AuEQyR5vVKQ/02iyzZZV7Se/pYs4OrVmRV8FW0R9KO77FcwCaCQTF+Nmp6AAAAABJRU5ErkJggg==)
Now, there are three cases from which the expected value of the amount which he wins can be calculated:
(i) First case is that, if he gets a six on his first through then the required probability will be ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAhCAMAAAAieUHKAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAADqQAGaQAGa2OgAAOjo6OmZmOmaQOma2OpC2OpDbZgA6ZjoAZjo6kDoAkNv/tmYAtmY625A625Bm27Zm29u229vb2////9uQ/9u2//+2///bmbpMdwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYklEQVQoU2NgwAAyQiwgMRF2XjDNwCCMn5YRZJYAqWIEAm5M43CIgFQDAdHqkRSK8nEALZQR4gJZC3OdDD8HHxtQXJKVU1yMn4VBmkcAKMfEIMPPKSED5DNI8TIycoJ1YAMAgu4DtSUphg0AAAAASUVORK5CYII=)
∴ Amount received by him = Rs. 1
(ii) Secondly, if he gets six on his second throw then the probability ![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ Amount received by him = - Rs. 1 + Rs. 1
= 0
(iii) Lastly, if he does not get six in first two throws and gets six in his third throw then the probability ![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ Amount received by him = - Rs. 1 – Rs. 1 + Rs. 1
= - 1
Hence, expected value that he can win ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 12.Suppose we have four boxes A, B, C and D containing coloured marbles as given below:
![](data:image/png;base64,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)
One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C?
Answer:Let us assume R be the event of drawing the red marbles
Let us also assume EA, EB andEC denote the boxes A, B and C respectively
It is given in the question that,
Total number of marbles = 40
Also, total number of red marbles = 15
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Probability of taking out the red marble from box A, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Also, probability of taking out the red marble from box B, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAqCAMAAACX8MsWAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOmaQOpDbZgAAZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bRdydOgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAq0lEQVQ4T71TSRLCMAyLWRqWsgQMBZrg//+yTuHAwQrDpIOP0USWbNm5KUvuO6LF2aZkOrjUzq8myo0+d3TEavoSyIA200WPWSVsYUcJWRMobh4Q65aK9TZxXOl0hG2QaSwsacpd/Y3r7UqNzUAyPqW8RpDri8DfaKvd3mzt0WepwFgpecVYVoFrD09QTnqCobSG6Avpe7b2RYxxRj9Z26UA7jPtdXqbS812BgWaBwX7vkeLAAAAAElFTkSuQmCC)
And, Probability of taking out the red marble from box C, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 13.Assume that the chances of a patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
Answer:Let us assume, X denotes the events having a person heart attack
A1 denote events having the selected person followed the course of yoga and meditation
And, A2 denote the events having the person adopted the drug prescription
It is given in the question that,
P (X) = 0.40
And, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ Probability (The patient suffering from a heart attack and followed a course of meditation and yoga):
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Question 14.If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability 1/2).
Answer:From the question, we have:
Total number of determinants of second order where the element being or 1 = (2)4
= 16
Now, we have the value of determinants is positive in following cases:
![](data:image/png;base64,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)
∴ Required probability = 3/16
Question 15.An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known:
P (A fails) = 0.2
P (B fails alone) = 0.15
P (A and B fail) = 0.15
Evaluate the following probabilities:
(i) P (A fails | B has failed)
(ii) P (A fails alone)
Answer:(i) Let us assume the event which is failed by A is denoted by EA
And, event which is failed by B is denoted by EB
It is given in the question that,
Event failed by A, P (EA) = 0.2
Event failed by both,![](data:image/png;base64,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)
And, event failed by B alone ![](data:image/png;base64,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)
0.15 = P (EB) – 0.15
∴ P (EB) = 0.30
Hence, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 0.5
(ii) We have,
Probability where A fails alone ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 16.Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.
Choose the correct answer in each of the following:
Answer:Let us firstly assume, A1 denote the events that a red ball is transferred from bag I to II
And, A2 denote the event that a black ball is transferred from bag I to II
∴ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFAAAAAgCAMAAACy7Q9eAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjpmZpDbZrbbZrb/kDoAkDo6kDqQkGYAkGY6kJA6kLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bSpZy1QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABh0lEQVRIS+WU2VLDMAxF7VCWspStbE2hNC7gQBKi//85LK9ZnFTM5IEZ9NJJqhxL90pm7B9Ffs6Txwn7hfWWfSbZhESFqk6nBX5vniYtUPDZABDeFh+Ro+TV+54CvuarWAakN0X0y+oimm9y67uMVRpYcozZ1jEgvfU4SI+a7FHR87kbG6nMhhfveHkQ+pW8BWSy/TiggERCyW07kJ74vPrhtU0gzYVpywONDiZ2K31YiKYcgwbVS9RMuC/LMO3VougAmQjlDwKxNthx50QAwjpjqG8zhJNAaCsxeusm8e2hNznUVKozGvVqrgf26rP4fo5H1MvI+cPA4OSyLYsHShTBeAU7l+M1HG656Sqe4p6rM/RXP+WX9xZoDBwP4QbQptk5tD+5MUtaIGEOlSUdm/SmwIYnqja1lvpfB6RtSqeD2PBaIKHAmByR28YAR28bSHEo4nPfuw9BHKvtHL8P4blguAqU0KO/314l+TUFR8+hFkgmloSbgwzDfSIqSGb++QLJnfwu8QdQ+yIPyyYvQQAAAABJRU5ErkJggg==)
And, ![](data:image/png;base64,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)
Let X be the event that the drawn ball is red
∴ when red ball is transferred from bag I to II, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
And, when black ball is transferred from bag I to II, ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAqCAMAAACX8MsWAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOmaQOpDbZgAAZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bRdydOgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAq0lEQVQ4T71TSRLCMAyLWRqWsgQMBZrg//+yTuHAwQrDpIOP0USWbNm5KUvuO6LF2aZkOrjUzq8myo0+d3TEavoSyIA200WPWSVsYUcJWRMobh4Q65aK9TZxXOl0hG2QaSwsacpd/Y3r7UqNzUAyPqW8RpDri8DfaKvd3mzt0WepwFgpecVYVoFrD09QTnqCobSG6Avpe7b2RYxxRj9Z26UA7jPtdXqbS812BgWaBwX7vkeLAAAAAElFTkSuQmCC)
Hence, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 17.If A and B are two events such that P(A) ≠ 0 and P(B | A) = 1, then
A. A ⊂ B
B. B ⊂ A
C. B = φ
D. A = φ
Answer:It is given in the question that,
A and B are two events where,
![](data:image/png;base64,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)
And, P (B|A) = 1
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
Hence, option A is correct
Question 18.If P (A|B) > P (A), then which of the following is correct:
A. P (B|A) < P (B)
B. P (A ∩ B) < P (A) . P (B)
C. P (B|A) > P (B)
D. P (B|A) = P (B)
Answer:It is given in the question that,
P (A|B) > P (A)
∴ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHEAAAAkCAMAAACJ4yWmAAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjpmZmaQZma2ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kJA6kLbbkNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm27aQ29v/2////7Zm/7aQ/9uQ/9u2//+2///b85PMNwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACiUlEQVRYR+VX2WLTMBC0kh4ulFAIZ4kptDZtVJcYY/3/r7GzWh12WhMffYF9SV0rGs3u7GqSJP9dPHy8e5rz/afdLAlp1kqpYwuk39otTXaKjzJViy/0yUsWl0m9mgdSn+wEwe+oz4FYLe+Sh3SDk5zsknK59SeaSDUnYjlzchTrVUkQJmMwvKEl5gf9Ub/YTgTD15v1RphYCOR0UxGh5t0NPYA3ZxUJN1/xr6lRp1QjVEtIMWGQad6DEBDx9OuMwNyZpmGiRk4vzLEiQmpxEyEis0jFPIgmE326/X6vL+3+niOWmILyPDCr5nYVl12/loYAHQkox2QK2lXLrUfUIP0Ky/uUw6lRR9c+4yZ7E7KHRNbnVidR/FWKTsxJ8Xm/MzUd3Vz580fJSzSxwIGhhXboi15NhAlgirOLn90vI++Vcjwgd4nmw3crlqAZ/93Dp5y5TaUuXnfY1SPKUOGXxUZb+EdI9lLsvDRlehyqRmKDBHPHrOZhxUGpEcQ400Ogwtry5RF3MgLsTMGDgh9DQWlsoMYIzDZW2IDoHM1tRTXCJoF0QKzoEO4p980/jmISc+xsJnmUi8e14DTEdh2bNV8GPjxH7ignqCmIXa1GSmnV0fa4e2vvqDGx34+5b0S7n2BIk9xbRVk9j4q9mUO6CdMSW1oo800tqEtoAvLbvn6MRsFIexPNHE+qPXNiv8NuwLmbkfbmkW7vUgx+RzCcuxlpb6K7Q0rbvTuC3xEI527G2psn7kef4+B35Kr37mbgRXywFIPfEUTvbuYxG/sHiXQUfCO7m2dCjKXFiMHdPFNWddzBaI7gbsYq5+CCYja0vbe3N0P2GLi2ZXjm+oHTf4bpU24gx39q+R8I+00jO+sp+wAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
P (B|A) > P (B)
Hence, option C is correct
Question 19.If A and B are any two events such that P(A) + P(B) – P(A and B) = P(A), then
A. P(B|A) = 1
B. P(A|B) = 1
C. P(B|A) = 0
D. P(A|B) = 0
Answer:It is given in the question that,
A and B are any two events where,
P (A) + P (B) – P (A and B) = P (A)
![](data:image/png;base64,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)
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∴ ![](data:image/png;base64,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)
= 1
Hence, option B is correct
A and B are two events such that P (A) ≠ 0. Find P(B|A), if:
(i) A is a subset of B
(ii) A ∩ B = φ
Answer:
It is given in the question that,
A and B are two events such that
We have,
∴
Hence,
= 1
(ii) We have,
∴
= 0
Question 2.
A couple has two children,
(i) Find the probability that both children are males, if it is known that at least one of the children is male.
(ii) Find the probability that both children are females, if it is known that the elder child is a female.
Answer:
(i) According to the situation, if the couple has two children then the sample space is:
S = {(b, b), (b, g), (g, b), (g, g)}
Let us assume A denote the event of both children having male and B denote the event of having at least one of the male children
Thus, we have:
Hence,
(ii) Let us now assume C denote the event having both children females and D denote the event of having elder child is female
∴ C = {(g, g)}
And, D = {(g, b), (g, g)}
Hence,
Question 3.
Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male?
Assume that there are equal number of males and females.
Answer:
It is given in the question that,
5% of men and 0.25% of women have grey hair
∴ Total % of people having grey hair = 5 + 0.25
= 5.25 %
Hence, Probability of having a selected person male having grey hair,
Question 4.
Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?
Answer:
It is given in the question that,
90% of the people are right handed
Let p denotes the probability of people that are right handed and q denotes the probability of people that are left handed
∴
And,
Now, by using the binomial distribution probability of having more than 6 right handed people can be given as:
Hence, the probability of having more than 6 right handed people:
= 1 – P (More than 6 people are right handed)
=
Question 5.
An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15 bear a mark 'Y'. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that:
(i) All will bear 'X' mark.
(ii) Not more than 2 will bear 'Y' mark.
(iii) At least one ball will bear 'Y' mark.
(iv) The number of balls with 'X' mark and 'Y' mark will be equal.
Answer:
(i) It is given in the question that,
Total number of balls in the urn = 25
Number of balls bearing mark ‘X’ = 10
Number of balls bearing mark ‘Y’ = 15
Let p denotes the probability of balls bearing mark ‘X’ and q denotes the probability of balls bearing mark ‘Y’
∴
And,
Now, 6 balls are drawn with replacement. Hence, the number of trials are Bernoulli triangle.
Let us assume, Z be the random variable that represents the number of balls bearing ‘Y’ mark in the trials
∴ Z has a binomial distribution where n = 6 and
Hence, P (All balls will bear mark ‘X’) = P (Z = 0)
(ii) Probability (Not more than 2 will bear ‘Y’ mark)
= P (Z = 0) + P (Z = 1) + P (Z = 2)
=
(iii) Now, Probability (At least one ball will bear ‘Y’ mark) = P (Z ≥ 1)
= 1 – P (Z = 0)
=
(iv) Probability (Having equal number of balls with ‘X’ mark and ‘Y’ mark) = P (Z = 3)
=
=
Question 6.
In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is 5/6. What is the probability that he will knock down fewer than 2 hurdles?
Answer:
In this question,
Let us assume p be the probability of player that will clear the hurdle while q be the probability of player that will knock down the hurdle
∴
And,
Let us also assume X be the random variable that represents the number of times the player will knock down the hurdle
∴ By binomial distribution,
Hence, probability (players knocking down less than 2 hurdles) = P (X < 2)
= P (X = 0) + P (X = 1)
=
=
=
Question 7.
A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.
Answer:
From the condition given in the question, it is clear that:
Probability of getting a six in a throw of die =
And, probability of not getting a six =
Let us assume,
Now, we have:
Probability that the 2 sixes come in the first five throws of the die
=
Also, Probability that the six come in the sixth throw
Question 8.
If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays?
Answer:
We know that, in a leap year there are total 366 days, 52 weeks and 2 days
Now, in 52 weeks there are total 52 Tuesdays
∴ Probability that the leap year will contain 53 Tuesdays is equal to the probability of remaining 2 days will be Tuesdays
Thus, the remaining two days can be:
(Monday and Tuesday), (Tuesday and Wednesday), (Wednesday and Thursday), (Thursday and Friday), (Friday and Saturday), (Saturday and Sunday) and (Sunday and Monday)
∴ Total Number of cases = 7
Cases in which Tuesday can come = 2
Hence, probability (leap year having 53 Tuesdays)
Question 9.
An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.
Answer:
It is given in the question that,
Probability of failure = x
And, probability of success = 2x
∴ x + 2x = 1
3x = 1
∴
Let us now assume
Also, X be the random variable that represents the number of trials
Hence, by binomial distribution we have:
∴ Probability of having at least 4 successes
= P (X = 4) + P (X = 5) + P (X = 6)
Question 10.
How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%?
Answer:
Let us assume that, man tosses the coin n times. Thus, n tosses are the Bernoulli trials
∴ Probability of getting head at the toss of the coin
Let us assume,
∴
It is given in the question that,
Probability of getting at least one head >
∴
1 – P (x = 0) > 0.9
(i)
Hence, the minimum value of n satisfying the given inequality = 4
∴ The man have to toss the coin 4 or more times
Question 11.
In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins / loses.
Answer:
For the situation given in the equation, we have:
Probability of getting a six in a throw of a die
Also, probability of not getting a 6
Now, there are three cases from which the expected value of the amount which he wins can be calculated:
(i) First case is that, if he gets a six on his first through then the required probability will be
∴ Amount received by him = Rs. 1
(ii) Secondly, if he gets six on his second throw then the probability
∴ Amount received by him = - Rs. 1 + Rs. 1
= 0
(iii) Lastly, if he does not get six in first two throws and gets six in his third throw then the probability
∴ Amount received by him = - Rs. 1 – Rs. 1 + Rs. 1
= - 1
Hence, expected value that he can win
Question 12.
Suppose we have four boxes A, B, C and D containing coloured marbles as given below:
One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C?
Answer:
Let us assume R be the event of drawing the red marbles
Let us also assume EA, EB andEC denote the boxes A, B and C respectively
It is given in the question that,
Total number of marbles = 40
Also, total number of red marbles = 15
∴
Probability of taking out the red marble from box A,
Also, probability of taking out the red marble from box B,
And, Probability of taking out the red marble from box C,
Question 13.
Assume that the chances of a patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
Answer:
Let us assume, X denotes the events having a person heart attack
A1 denote events having the selected person followed the course of yoga and meditation
And, A2 denote the events having the person adopted the drug prescription
It is given in the question that,
P (X) = 0.40
And,
∴ Probability (The patient suffering from a heart attack and followed a course of meditation and yoga):
=
Question 14.
If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability 1/2).
Answer:
From the question, we have:
Total number of determinants of second order where the element being or 1 = (2)4
= 16
Now, we have the value of determinants is positive in following cases:
∴ Required probability = 3/16
Question 15.
An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known:
P (A fails) = 0.2
P (B fails alone) = 0.15
P (A and B fail) = 0.15
Evaluate the following probabilities:
(i) P (A fails | B has failed)
(ii) P (A fails alone)
Answer:
(i) Let us assume the event which is failed by A is denoted by EA
And, event which is failed by B is denoted by EB
It is given in the question that,
Event failed by A, P (EA) = 0.2
Event failed by both,
And, event failed by B alone
0.15 = P (EB) – 0.15
∴ P (EB) = 0.30
Hence,
= 0.5
(ii) We have,
Probability where A fails alone
Question 16.
Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.
Choose the correct answer in each of the following:
Answer:
Let us firstly assume, A1 denote the events that a red ball is transferred from bag I to II
And, A2 denote the event that a black ball is transferred from bag I to II
∴
And,
Let X be the event that the drawn ball is red
∴ when red ball is transferred from bag I to II,
And, when black ball is transferred from bag I to II,
Hence,
Question 17.
If A and B are two events such that P(A) ≠ 0 and P(B | A) = 1, then
A. A ⊂ B
B. B ⊂ A
C. B = φ
D. A = φ
Answer:
It is given in the question that,
A and B are two events where,
And, P (B|A) = 1
∴
∴
Hence, option A is correct
Question 18.
If P (A|B) > P (A), then which of the following is correct:
A. P (B|A) < P (B)
B. P (A ∩ B) < P (A) . P (B)
C. P (B|A) > P (B)
D. P (B|A) = P (B)
Answer:
It is given in the question that,
P (A|B) > P (A)
∴
P (B|A) > P (B)
Hence, option C is correct
Question 19.
If A and B are any two events such that P(A) + P(B) – P(A and B) = P(A), then
A. P(B|A) = 1
B. P(A|B) = 1
C. P(B|A) = 0
D. P(A|B) = 0
Answer:
It is given in the question that,
A and B are any two events where,
P (A) + P (B) – P (A and B) = P (A)
∴
= 1
Hence, option B is correct