Probability Class 12th Mathematics Part Ii CBSE Solution

Class 12th Mathematics Part Ii CBSE Solution
Exercise 13.1
  1. Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) =…
  2. Compute P(A|B), if P(B) = 0.5 and P (A ∩ B) = 0.32
  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)…
  4. Evaluate P(A ∪ B), if 2P(A) = P(B) = 5/13 and P(A|B) = 2/5.
  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)…
  6. A coin is tossed three times, where (i) E : head on third toss, F : heads on…
  7. Two coins are tossed once, where (i) E : tail appears on one coin, F : one coin…
  8. A die is thrown three times, E : 4 appears on the third toss, F : 6 and 5…
  9. Mother, father and son line up at random for a family picture E : son on one…
  10. A black and a red dice are rolled. (a) Find the conditional probability of…
  11. A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5}…
  12. Assume that each born child is equally likely to be a boy or a girl. If a…
  13. An instructor has a question bank consisting of 300 easy True / False…
  14. Given that the two numbers appearing on throwing two dice are different. Find…
  15. Consider the experiment of throwing a die, if a multiple of 3 comes up, throw…
  16. If P (A) = 1/2, P(B) = 0, then P (A|B) isA. 0 B. 1/2 C. not defined D. 1…
  17. 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
  1. If P(A) = 3/5 and P (B) = 1/5, find P (A ∩ B) if A and B are independent…
  2. Two cards are drawn at random and without replacement from a pack of 52 playing…
  3. A box of oranges is inspected by examining three randomly selected oranges…
  4. A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on…
  5. A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event,…
  6. Let E and F be events with P (E)= 3/5, P (F) = 3/10 and P (E ∩ F) = 1/5. Are E…
  7. Given that the events A and B are such that P(A) = 1/2, P (A ∪ B) = 3/5 and…
  8. Let A and B be independent events with P (A) = 0.3 and P(B) = 0.4. Find (i) P(A…
  9. If A and B are two events such that P (A) = 1/4 , P (B) = 1/2 and P (A ∩ B) =…
  10. Events A and B are such that P (A) = 1/2, P(B) = 7/12 and P(not A or not B) =…
  11. Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6. Find…
  12. A die is tossed thrice. Find the probability of getting an odd number at least…
  13. Two balls are drawn at random with replacement from a box containing 10 black…
  14. Probability of solving specific problem independently by A and B are 1/2 and…
  15. One card is drawn at random from a well shuffled deck of 52 cards. In which of…
  16. In a hostel, 60% of the students read Hindi newspaper, 40% read English…
  17. The probability of obtaining an even prime number on each die, when a pair of…
  18. Two events A and B will be independent, if(A) A and B are mutually exclusive…
Exercise 13.3
  1. An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour…
  2. A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black…
  3. Of the students in a college, it is known that 60% reside in hostel and 40% are…
  4. In answering a question on a multiple choice test, a student either knows the…
  5. A laboratory blood test is 99% effective in detecting a certain disease when it…
  6. There are three coins. One is a two headed coin (having head on both faces),…
  7. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000…
  8. A factory has two machines A and B. Past record shows that machine A produced…
  9. Two groups are competing for the position on the Board of directors of a…
  10. Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three…
  11. A manufacturer has three machine operators A, B and C. The first operator A…
  12. A card from a pack of 52 cards is lost. From the remaining cards of the pack,…
  13. Probability that A speaks truth is 4/5. A coin is tossed. A reports that a…
  14. If A and B are two events such that A ⊂ B and P(B) ≠ 0, then which of the…
Exercise 13.4
  1. X 0 1 2 P(X) 0.4 0.4 0.2 State which of the following are not the probability…
  2. 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…
  3. Y -1 0 1 P(Y) 0.6 0.1 0.2 State which of the following are not the probability…
  4. 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…
  5. An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X…
  6. Let X represent the difference between the number of heads and the number of…
  7. number of heads in two tosses of a coin. Find the probability distribution of…
  8. number of tails in the simultaneous tosses of three coins. Find the…
  9. number of heads in four tosses of a coin. Find the probability distribution of…
  10. Find the probability distribution of the number of successes in two tosses of a…
  11. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn…
  12. A coin is biased so that the head is 3 times as likely to occur as tail. If the…
  13. A random variable X has the following probability distribution: X 0 1 2 3 4 5 6…
  14. The random variable X has a probability distribution P(X) of the following…
  15. Find the mean number of heads in three tosses of a fair coin.
  16. Two dice are thrown simultaneously. If X denotes the number of sixes, find the…
  17. Two numbers are selected at random (without replacement) from the first six…
  18. Let X denote the sum of the numbers obtained when two fair dice are rolled.…
  19. A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18,…
  20. In a meeting, 70% of the members favour and 30% oppose a certain proposal. A…
  21. The mean of the numbers obtained on throwing a die having written 1 on three…
  22. Suppose that two cards are drawn at random from a deck of cards. Let X be the…
Exercise 13.5
  1. A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the…
  2. A pair of dice is thrown 4 times. If getting a doublet is considered a success,…
  3. There are 5% defective items in a large bulk of items. What is the probability…
  4. Five cards are drawn successively with replacement from a well-shuffled deck of…
  5. The probability that a bulb produced by a factory will fuse after 150 days of…
  6. A bag consists of 10 balls each marked with one of the digits 0 to 9. If four…
  7. In an examination, 20 questions of true-false type are asked. Suppose a student…
  8. Suppose X has a binomial distribution b (6 , 1/2) . Show that X = 3 is the most…
  9. On a multiple choice examination with three possible answers for each of the…
  10. A person buys a lottery ticket in 50 lotteries, in each of which his chance of…
  11. Find the probability of getting 5 exactly twice in 7 throws of a die.…
  12. Find the probability of throwing at most 2 sixes in 6 throws of a single die.…
  13. It is known that 10% of certain articles manufactured are defective. What is…
  14. In a box containing 100 bulbs, 10 are defective. The probability that out of a…
  15. The probability that a student is not a swimmer is 1/5. Then the probability…
Miscellaneous Exercise
  1. A and B are two events such that P (A) ≠ 0. Find P(B|A), if: (i) A is a subset…
  2. A couple has two children, (i) Find the probability that both children are…
  3. Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person…
  4. Suppose that 90% of people are right-handed. What is the probability that at…
  5. An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15…
  6. In a hurdle race, a player has to cross 10 hurdles. The probability that he…
  7. A die is thrown again and again until three sixes are obtained. Find the…
  8. If a leap year is selected at random, what is the chance that it will contain…
  9. An experiment succeeds twice as often as it fails. Find the probability that in…
  10. How many times must a man toss a fair coin so that the probability of having…
  11. In a game, a man wins a rupee for a six and loses a rupee for any other number…
  12. Suppose we have four boxes A, B, C and D containing coloured marbles as given…
  13. Assume that the chances of a patient having a heart attack are 40%. It is also…
  14. If each element of a second order determinant is either zero or one, what is…
  15. An electronic assembly consists of two subsystems, say, A and B. From previous…
  16. Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black…
  17. If A and B are two events such that P(A) ≠ 0 and P(B | A) = 1, thenA. A ⊂ B B.…
  18. If P (A|B) P (A), then which of the following is correct:A. P (B|A) P (B) B. P…
  19. 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,




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:

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) 


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.


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) 


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)05C1(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