Q. 5 What is conditional probability? Explain with an example.

Ans. C O N D I T I O N A L P R O B A B I L I T Y

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the (conditional) probability of A, given B" or "the probability of A under the condition B". When in a random experiment the event B is known to have occurred, the possible outcomes of the experiment are reduced to B, and hence the probability of the occurrence of A is changed from the unconditional probability into the conditional probability given B.

Two events A and B are said to be dependent when B can occur only when A is known to have occurred or vice versa. The probabilities associated with such event are called conditional probabilities.

The probabilities of the occurrence of the event A when the event B has already occurred is called the conditional probability of occurrence of A given that the event B has already occurrence and is denoted by P(A/B).

Example:- A pair of dice is rolled. If the sum of 9 has appeared, find the probability that one of the dice shows 3.

Solution:- The equiprobable sample space consisting of 36 sample points.

The event A = The sum of the scores is 9 has four sample points (6,3), (5,4), (4,5), (3,6), and its reduced sample space.

Under the assumption that A has happened, the event B = one of the dice shows 3 has only two sample points, that is (B∩A) = [(3,6),(6,3)].

∴ P(B/A) = 2/4 = 1/2

Also using the formula derived above, we get P(B/A) = P (A ∩B)P(A )

[∴P (A∩B )=n(A∩B)n(S)

= 236,∧P (A )=n (A )

n(S)= 436 ]

¿ 2/364 /36

=24=1 /2