[Math Lair] Monty Hall Problem

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The Monty Hall problem is a problem related to probability. It is named after the host of the game show Let's Make a Deal, from which the problem was inspired. The problem was first posed in 1975. The problem goes as follows:

You are on a game show. You are given the choice of three boxes, A, B, and C. In one of the boxes are the keys to a new luxury car; the other two boxes are empty. You pick one of the boxes, say box B, and the host, who knows what's in the boxes and is not going to open the box with the keys in it, opens another box, say box A, which is empty. He then asks you whether you'd like to switch your box for box C. Should you switch, or not?

This problem became famous in 1990 when Marilyn vos Savant featured it in her column in Parade magazine. She gave the answer that the sticking to your original choice has a ⅓ chance of winning, but switching gives a ⅔ chance of winning. Vos Savant received thousands of letters about this problem, the vast majority of which, including a majority of letters sent by academics, disagreed with her. However, after several further explanations, most people realized that vos Savant's solution was correct.

There are a few ways of looking at this problem. Here are two:

  1. If you pick an empty box to start, Monty will then open the other empty box. In this case, you will win by switching. If you originally picked the box with the keys in it, you will lose by switching. So, the probability that you will win by switching is the same as the probability that you will pick the wrong box to begin with, which is ⅔.
  2. At the start, you have no reason to believe that any box is more or less likely than any other to be the winner, so the probability of each box having the keys in it is ⅓. After you pick your box, Monty shows you an empty box. Since he can always do so, this doesn't change the ⅓ probability that your selected box contains the keys. Since there is only one other unopened box, that box must have a probability of ⅔ of containing the keys behind it, so the probability of winning if you switch to that box must be ⅔.

Sources used (see bibliography page for titles corresponding to numbers): 47.