3. The prior probabilities for events A₁, A₂, and A₃ are P(A₁) = .30, P(A₂) = .50, and P(A₃) = .20. The conditional probabilities of event B given A₁, A₂, and A₃, are P(B|A₁) = .30, P(B|A₂) = .40, and P(B|A₃) = .50.

a. Compute P(BA₁), P(BA₂), and P(BA₃).

b. Apply Bayes' theorem, equation (4.19), to compute the posterior probability P(A₃|B).

c. Use the tabular approach to applying Bayes' theorem to compute P(A₁|B), P(A₂|B), and P(A₃|B).