# Bayes’ Theorem

Bayes’ theorem follows easily from the rules of probability but is listed separately here because of its special usefulness.

THEOREM 2.4.2 (Bayes) Let events A, A2, . . . , An be mutually exclusive such that P{A U A2 U. . . U An) = 1 and Р(Д) > 0 for each i. Let E be an arbitrary event such that P(E) > 0. Then

Proof. From Theorem 2.4.1, we have

Since E П A], E П А4, . . . , E fl An are mutually exclusive and their union is equal to E, we have, from axiom (3) of probability,

(2.4.4) P(E) = X P(E n Aj)-

j= і

Thus the theorem follows from (2.4.3) and (2.4.4) and by noting that P(E П Aj) = P(£ I Aj)P(Aj) by Theorem 2.4.1. □

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