Chapter 4.5, Problem 40E

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

1. a. Compute P(B ∩ A₁), P(B ∩ A₂), and P(B ∩ A₃).
2. b. Apply Bayes’ theorem, equation (4.19), to compute the posterior probability P(A₂ ∣ B).
3. c. Use the tabular approach to applying Bayes’ theorem to compute P(A₁ ∣ B), P(A₂ ∣ B), and P(A₃ ∣ B).