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.