Conditional Probability

Scott E. Brodie

5/22/1999

Consider an "event", A, and a set of events, B_i, where the B_i are "mutually exclusive" (no two can occur simultaneously) and "exhaustive" (one of the B_i always occurs). Then the event that both A and B_i occur together is given by the intersection . Since no two of the B_i  can occur together, the events  are likewise mutually exclusive, and the event A can be recovered by taking the union of the events . We can therefore express the probability of A in terms of the probabilities of the joint events :

If none of the P(B_i) are zero, we may write this sum in the form

         (*).

The quantity is known as the "conditional probability of A given B_i", as it represents the fraction of the likelihood of the event B_i during which the event A simultaneously occurs. Often, by making use of the additional information as to which of the B_i has occurred, it is easier to determine the conditional probabilities  than the joint probabilities P(). If the probabilities P( B_i) are known, then (*) can be used to recover the probability  P(A).