Probability

Introduction : Probability is a branch of mathematics which mainly means 'the calculation of probable chances.' For example, if you flip a coin, chances of getting heads or tails is equally 1/2. It means that, as there are two options for the flip, the chances of getting one of the two options is always 1/2. Similarly you have 1/6 chances of getting 1 on a die (same for all other numbers) and 1/52 chances of getting an Ace of Spades out of the 52 playing cards.

As the possible options increase, chances decrease as 1/2 is always more possible than 1/52. You would get more possible chances of getting a black pebble out of a box containing 3 black and 4 white pebbles than out of a box containing 1 black and 3 white pebbles. (3/4 > 1/3)

The possible chances are called outcomes. A given no. of outcomes is equally likely if none of them occur in preference to the others. The set of all possible outcomes of a random experiment is calledsample space and is denoted by 'S'.
 Sr. No. Random Experiments Sample Space No. of sample points 
 1A coin is tossed S = {H,T} n(S) = 2 
 2Two coins are tossed S = {HH, HT, TH, TT} n(S) = 4 
Three coins are tossed S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT  n(S) = 8 
 4A die is thrown S = {1,2,3,4,5,6}  n(S) = 6
 A card is drawn from a pack containing 25 cards S = {1,2,3, ... ,25}  n(S) = 25
Outcomes satisfying the particular condition are called favorable outcomes. A set of all favorable outcomes is called an event

Types of Events :
Certain Event : An event which contains all sample points of a sample space is called a certain event. e.g. A die is thrown. Let M be the event of getting a score less than 10 and so M = {1,2,3,4,5,6} and hence n(M) = 6.
Impossible Event : An event which does not contain any sample points of a sample space is called an impossible event. e.g. A die is thrown. Let A be the event of getting a score divisible by 7 and so S = {1,2,3,4,5,6} and A = {} and hence n(A) = 0.
Elementary Event : An event containing only one sample point of a sample space is called an elementary event. e.g. A die is thrown. Let A be the event of getting a score divisible by 5 and so S = {1,2,3,4,5,6} and A = {5} and hence n(A) = 1.
Complementary Event : If A is an event of a sample space S the the set of all outcomes which are in S but not in A is called the complement of event A and denoted by A'. A and A' are complementary events.
Mutually Exclusive Events : If A and B are two events of a sample space S and if they do not have any sample point is common, then they are called mutually exclusive events. 
Exhaustive Events : If the union of two or more events is a sample space then those events are exhaustive events.

Probability of an Event : Denoted as P(A), the probability of an event is P(A) = n(A) / n(S)

Properties of Probability :
  • Probability of an impossible event is zero and that of a certain event is 1.
  • If S is the finite sample space and A is an event of S then 0 <= P(A) <= 1.
  • If S is a finite sample space and A is an event of S then P(A') = 1 - P(A) where A' is the complement of event A.