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. At long last the truth was established and accepted. Below is one simulation you may try on your computer. For simplicity, I do not hide goats behind the doors. There is only one 'abstract' prize. You may either hit on the right door or miss it. You make your selection by pressing small round buttons below input controls that substitute for the doors. Down below other controls update experiment statistics even as you progress. Approximately every second the program clears 'door' controls and is waiting for your selection. Before you start, set up a desirable total number of experiments. With every selection it will decrease by 1. Two controversial solutions are given after the puzzle. Which is the right one? Important note:You run a simulation. During a simulation you are allowed to make as many selections as indicated in the "To Go" control before your first selection. Remember also that after each selection the device needs approximately 1 second to clear up controls. Please wait till it does. To start a new simulation please press the "Reset" button. 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 have 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 cansiters to simulate doors in the stage performance.)
Terry Pascal offered his variant of the solution.
"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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