# How to evaluate P(A/B) or P(B/A)

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How to evaluate P(A/B) or P(B/A).

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If the event A occurs when B has already occurred, then P(B) ≠ 0, then we may regard B as a new (reduced) sample space for event A.

In that case, the outcomes favourable to the occurrence of event A are those outcomes which are favourable to B as well as favourable to A, i.e, the outcomes favourable to A ∩ B and probability of occurrence of A so obtained is the conditional probability of A under the condition that B has already occurred.

P(A/B) = $\frac{Number\,of\,outcomes\,favourable\,to\,both\,A\,and\,B}{Number\,of\,outcomes\,in\,sample\,space(B,here)}$

=$\frac{n(A\,\cap\,B)}{n(B)}$ =   $\frac{\frac{n(A\,\cap\,B)}{n(S)}}{\frac{n(B)}{n(S)}}$ = $\frac{P(A\,\cap\,B)}{P(B)}$ ,  where S is the sample space for the events A and B.

Similarly,

P(B/A) = $\frac{P(A\,\cap\,B)}{P(A)}$ , P(A) ≠ 0

where P(B/A) is the conditional probability of occurrence of B, knowing that A has already occurred.

Note: If A and B are mutually exclusive events, then,

P(A/B) = $\frac{P(A\,\cap\,B)}{P(B)}$ = 0 ∵ $P(A\,\cap\,B)$ = 0

P(B/A) =$\frac{P(A\,\cap\,B)}{P(A)}$ =0 ∵ $P(A\,\cap\,B)$ =0