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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.


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

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