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Bayes' Theorem ) ~%

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S352 Theorem of total probability v, Ww ~—, Let {E,,E,,...,.E,} be a partition of the sample space S, and suppose that each of the, events E, , E,,..., E, has nonzero probability of occurrence. Let A be any event associated, with S, thei = A SS, P(A) = P(E,) P(AIE,) + P(E,) P(AIE,) + ... + P(E) PALE, |, , —_, , , , , = )'PE,)P(AIE,), j=l, , a Q”, $: En& ja — fu éu mse, LU, , P (ei) Po”

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ayes’ Theorem If E,, E, ,..., E, are n non empty events which constitute a partition, of sample space S, i.e. E,, E,,..., E, are pairwise disjoint and EU E,U ... UE, =S and, Ais any event of nonzero probability, then, , P(E,)P(AIE,), , , , , , P(EJA) = for any i= 1, 2, 3, ....”, , YP, )P(AE,, a

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Remark The following terminology is generally used when Bayes' theorem is applied., % The events E,, E. sag EB, are calle, 7 The probability PQ PE) i is s called the pen probability of the the hypothesi, , o The conditional p probability | PE, IA) is called a posterior p probability of the, hypothesis E,. ee a, , wre