## 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 Two controversial solutions are given after the puzzle. Which is the right one?
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 #2
After 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
The above simulation tool have the virtue of being quite suggestive - three quantities, viz., - the number of hits
- the number of wins with no switching
- the number of losses with switching
are all equal. It's really better to see once... Multi-Stage Monty Hall Dilemma
"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:
"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 Three Shell Game
Martin Gardner in his
After playing the game a while, Mr. Mark decided he couldn't win more than once out of three.
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? Comment
The 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- A.K.Dewdney,
*200% of Nothing*, John Wiley & Sons, Inc., 1993 - Martin Gardner,
*aha! Gotcha. Paradoxes to puzzle and delight*, Freeman & Co, NY, 1982 - Marylin vos Savant,
*The Power of Logical Thinking*, St. Martin's Press, NY 1996
## On Internet- http://math.rice.edu/~ddonovan/montyurl.html">The WWW Tackles The Monty Hall Problem
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