# Conditional Probability Theorem

Consider two events A and B defined on a sample space S. The probability of occurrence of event A given that event B has already occurred is known as the conditional probability of A relative to B.

It implies that the outcomes favorable to B become the possible outcomes and hence outcomes favorable to ) are outcomes common to A and B.

Let  and. Suppose that the event B occurs. Then there are exactly a sample points and these points can be considered to be the sample space for the other event A. The event A in this sample space consists of b sample points common to A and B.

Therefore, the probability of A in this sample space = b/a.

Thus the probability of A under the assumption that event B has occurred is :

P (A/B) = = =

=>                       P (A/B) =

and similarly =>  P (B/A) =

The above result is known as conditional probability theorem.