The prior probabilities for events Aj , A2, and A3 are P(A, ) = 0.20, P(A2) = 0.30, and P(A;) = 0.50. The conditional probabilities of event B given A1, A2, and A3 are P(B | A, ) = 0.40, P(B | A2) = 0.20, and P(B | A3) = 0.50. (Assume that Aj , A2, and A3 are mutually exclusive events whose union is the entire sample space.) (a) Compute P(B n A,), P(B n A2), and P(B n A3). P(B n A] ) = P(B n A2) = P(B n A3) P(A )P(B I A,) (b) Apply Bayes' theorem, P(A; | B) = P(A, )P(B | A, ) + P(A2) P(B | A2) + .. . + P(A,,)P(B IA)' to compute the posterior probability P(Az | B). (Round your answer to two decimal places.) (c) Use the tabular approach to applying Bayes' theorem to compute P(A, | B), P(A2 | B), and P(A | B). ( Round your answers to two decimal places.) Events P(A;) P(B | A;) P(A; n B) P(A; I B) A1 0.20 0.40 A2 0.30 0.20 A3 0.50 0.50 1.00 1.00