Consider the three prisoners scenario described in Example 5.26. Let A, B, and C denote respectively the events that Alice is to be released, Bob is to be released, and Carl is to be released, which we assume to be equally likely, so Pr(A) Pr(B) Pr(C) = 름. Also let J be the event that the jailer tells Aice that Bob is to stay in jail. (a) Compute the values of Pr(B | J), Pr(J| B), and Pr(J C) (b) Compute the values of Pr(J A) and Pr(J | Ac), where the event A is the event that Alice stays in jail. (c) Suppose that if Alice is the one who is to be released, then the jailer flips a fair coin to decide whether to tell Alice that Bob stays in jail or that Carl stays in jail. What is the value of Pr(A |J)? d) Suppose instead that if Alice is the one who is to be released, then the jailer always tells her that Bob will stay in jail. Now what is the value of Pr(AIJ)? Other similar problems with counterintuitive conclusions include the Monty Hall problem (Exercise 5.27), Bertrand's box paradox, and the principle of restricted choice in contract bridge.
