In this exercise we will use Bayes' theorem to solve the Monty Hall puzzle (Example 10 in Section 7.1). Recall that in this puzzle you are asked to select one of three doors to open. There is a large prize behind one of the three doors and the other two doors are losers. After you select a door, Monty Hall opens one of the two doors you did not select that he knows is a losing door, selecting at random if both are losing doors. Monty asks you whether you would like to switch doors. Suppose that the three doors in the puzzle are labeled 1, 2, and 3. Let W be the random variable whose value is the number of the winning door; assume that p(W = k) = 1/3 for k = 1, 2, 3. Let M denote the random variable whose value is the number of the door that Monty opens. Suppose you choose door i.
a) What is the probability that you will win the prize if the game ends without Monty asking you whether you want to change doors?
b) Find p(M = j | W = k) for j = 1, 2, 3 and k = 1, 2, 3.
c) Use Bayes' theorem to find p(W = j | M = k) where i and j and k are distinct values.
d) Explain why the answer to part (c) tells you whether you should change doors when Monty gives you the chance to do so.