1.3 Cars and goats: the Monty Hall dilemma On Sunday September 9, 1990, the following question...

1.3 Cars and goats: the Monty Hall dilemma On Sunday September 9, 1990, the following question appeared in the "Ask Marilyn" column in Parade, a Sunday supplement to many newspapers across the United States: Suppose you're on a game show, and you're given the choice of three doors; behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?-craig F. Whitaker, Columbia, Md Marilyn's answer-one should switch-caused an avalanche of reactions, in total an estimated 10000 Some of these reactions were not so flattering ("You are the goat"], quite a lot were by professional mathematicians ("You blew it, and blew it big," "You are utterly incorrect... How many irate mathematicians are needed to change your mind?"). Perhaps some of the reactions were so strong, because Marilyn vos Savant, the author of the column, is in the Guinness Book of Records for having one of the highest IQs in the world The switching question was inspired by Monty Hall's "Let's Make a Deal" game show, which ran with small interruptions for 23 years on various U.S. television networks. Although it is not explicitly stated in the question, the game show host will always open a door with a goat after you make your initial choice Many people would argue that in this situation it does not matter whether one would change or not: one door has a car behind it, the other a goat, so the odds to get the car are fifty-fifty. To see why they are wrong, consider the following argument. In the original situation two of the three doors have a goat behind them, so with probability 2/3 your initial choice was wrong, and with probability 1/3 it was right. Now the host opens a door with a goat (note that he can always do this). In case your initial choice was wrong the host has only one option to show a door with a goat, and switching leads you to the door with the car. In case your initial choice was right the host has two goats to choose from, so switching will lead you to a goat. We see that switching is the best strategy, doubling our chances to win. To stress this argument, consider the following generalization of the problem: suppose there are 10000 doors, behind one is a car and behind the rest, goats. After you make your choice, the host will open 9998 doors with goats, and offers you the option to switch. To change or not to change, that's the question! Still not convinced? Use your Internet browser to find one of the zillion sites where one can run a simulation of the Monty Hall problem (more about simulation in Chapter 6). In fact, there are quite a lot of variations on the problem. For example, the situation that there are four doors: you select a door, the host always opens a door with a goat, and offers you to select another door. After you have made up your mind he opens a door with a goat, and again offers you to switch. After you have decided, he opens the door you selected. What is now the best strategy? In this situation switching only at the last possible moment yields a probability of 3/4 to bring the car home. Using the law of total probability from Section 3.3 you will find that this is indeed the best possible strategy

3.14 Recall the Monty Hall problem from Section 1.3. Let R be the event "the prize is behind the door you chose initially," and W the event "you win the prize by switching doors." a. Compute P(W | R) and P(W | R) b. Compute P(W) using the law of total probability.