## Conditional Probability

Scott E. Brodie

5/22/1999

Consider an "event", *A*, and a set of events, *B _{i}*, where the

*B*are "mutually exclusive" (no two can occur simultaneously) and "exhaustive" (one of the

_{i }*B*always occurs). Then the event that both

_{i }*A*and

*B*occur together is given by the intersection . Since no two of the

_{i }*B*can occur together, the events are likewise mutually exclusive, and the event

_{i }*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*during which the event

_{i }*A*simultaneously occurs. Often, by making use of the additional information as to which of the

*B*has occurred, it is easier to determine the conditional probabilities than the joint probabilities

_{i}*P*(). If the probabilities

*P*(

*B*) are known, then (*) can be used to recover the probability

_{i}*P*(

*A*).

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

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