Conditional Probability

Scott E. Brodie

5/22/1999

Consider an "event", A, and a set of events, Bi, where the Bi are "mutually exclusive" (no two can occur simultaneously) and "exhaustive" (one of the Bi always occurs). Then the event that both A and Bi occur together is given by the intersection . Since no two of the B 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(Bi) are zero, we may write this sum in the form

(*).

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

(For an additional perspective, see another Conditional Probability page.)