Monty Hall DilemmaThe Monty Hall Dilemma was discussed in the popular "Ask Marylin" question-and-answer column of the Parade magazine. Details can also be found in the "Power of Logical Thinking" by Marylin vos Savant, St. Martin's Press, 1996. Marylin received the following question:
Marylin's response caused an avalanche of correspondence, mostly from people who would not accept her solution. Several iterations of correspondence ensued. Eventually, she issued a call to Math teachers among her readers to organize experiments and send her the charts. Some readers with access to computers ran computer simulations. Years ago I concluded this paragraph with the sentence, At long last, the truth was established and accepted. In fact truth proved to be more complex. With the accumulated experience, it is safe to assert that the major thrust of the controversy was not directed at Marylin's solution but at her interpretation of Craig's formulation. Indeed, there are two ways to interpret Monty's behavior as described by Craig Whitaker, "... and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat." One interpretation stems from the disregard of the piece of information conveyed by the phrase "the host, who knows what's behind the doors." According to this interpretation, Craig's statement nowhere says that the host acted on his knowledge of what's behind the doors. He just opens the door behind which (apparently by pure chance) is found a goat. The only reason to mention his knowledge one may think is to avoid the need to mention his lack of surprise at the find. During the whole operation the host remained as nonchalant as he is supposed to be. Marylin, on the other hand, chose another interpretation. She sensed that there is a reason that host's knowledge of the location of the items behind the doors has been mentioned explicitly. And although it was not followed by an equally explicit statement to the effect that, based on his knowledge, the host always opens the door to reveal a goat, this is what the problem was about. For more on the controversy and its history please check the Monty Hall Problem article at the wikipedia. There are two simulations based on Marylin's interpretation. One is plain and another with a twist that speeds up the simulation progress and, perhaps adds insight on what's happening with Monty's Dilemma.  Solution #1 There is a 1/3 chance that you'll hit the prize door, and a 2/3 chance that you'll miss the prize. If you do not switch, 1/3 is your probability to get the prize. However, if you missed (and this with the probability of 2/3) then the prize is behind one of the remaining two doors. Furthermore, of these two, the host will open the empty one, leaving the prize door closed. Therefore, if you miss and then switch, you are certain to get the prize. Summing up, if you do not switch your chance of winning is 1/3 whereas if you do switch your chance of winning is 2/3. Solution #2After the host opened one door, two remained closed with equal probabilities of having the prize behind them. Therefore, regardless of whether you switch or not you have a 50-50 chance(i.e, with probabilities 1/2) to hit or miss the prize door. Remark 1The above simulation tool has the virtue of being quite suggestive - three quantities, viz.,
are all equal. It's really better to see once... Remark 2S. K. Stein in his book Strength in Numbers makes use of the Monty Hall Dilemma to demonstrate a mathematician's approach to problem solving. First run 50 experiments. Next think of the results. (In the following he uses 35 mm film canisters to simulate doors in the stage performance.)
Remark 3Another solution can be obtained via the Principle of Proportionality. Terry Pascal offered his variant of the solution. Also check Ashutosh Joshi's description of an approach that helped him come to terms with Marylin's solution. Bruno Barros found a different approach. Yet another view has been presented by Peter Stikker. One of the solutions that come naturally to me has been sent by Michael Gerard Wilson:
Kenneth Kaplan found that counting the losing probabilities helps arrive at the right conclusion:
A similar argument with a credit to Erich Neuwirth has been published in The College Mathematical Journal (vol 30, no 5, November 1999, p. 369. Erich put it this way:
Keith Devlin has come with a new twist to the problem:
Marcus Bizony came up with a device to help overcome a psychological barrier many encounter pondering the problem:
"In the three-door Monty Hall Dilemma, there are two stages to the decision, the initial pick followed by the decision to stick with it or switch to the only other remaining alternative after the host has shown an incorrect door. An intriguing extension of the basic Monty Hall Dilemma has been provided by M. Bhaskara Rao of the Department of Statistics at the North Dakota University. He analyzed what happens when the dilemma is expanded beyond the two stages. The number of stages can be as many as the number of doors minus one. "Suppose there are four doors, one of which is a winner. The host says: "You point to one of the doors, and then I will open one of the other non-winners. Then you decide whether to stick with your original pick or switch to one of the remaining doors. Then I will open another (other than the current pick) non-winner. You will then make your final decision by sticking with the door picked on the previous decision or by switching to the only other remaining door. "Now there are three stages, and the four different strategies can be summarized as follows:
"People who accept the correctness of the 2/3 solution in the basic Monty Hall Dilemma might assume that one does best by switching in both Stage 2 and Stage 3. However, as shown here, the counter-intuitive solution to the three-stage Monty Hall Dilemma is to stick in Stage 2 and to switch in Stage 3. These remarkable probabilities were published by Rao in the American Statistician. The underlying principle is that in a multi-stage Monty Hall Dilemma, one should stick with one's initial hunch until the very last chance and then switch."
Martin Gardner in his Aha! Gotcha describes the following variant: Operator: Step right up, folks. See if you can guess which shell the pea is under. Double your money if you win. After playing the game a while, Mr. Mark decided he couldn't win more than once out of three. Operator: Don't leave, Mac. I'll give you a break. Pick any shell. I'll turn over an empty one. Then the pea has to be under one of the other two, so your chances of winning go way up. Poor Mr. Mark went broke fast. He did not realize that turning an empty shell had no effect on his chances. Do you see why? CommentThe problem is actually the same but looked at from a different perspective. Since Mr. Mark has made his choice no Operator's action can change his chances. So, to me at least, the Shell Game makes it pretty obvious that unless you switch in the Monty Hall Dilemma (i.e. if you play the Shell Game), you chances remain 1 to 3. However, if you switch, you select one door out of two.
References
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