During the late 1990s, Marilyn vos Savant, holder of the *Guinness Book of Records’* highest recorded IQ of 228, received an avalanche of hostile responses, many from Ph.D.s in math and statistics, when she correctly solved the controversial “Monty Hall Problem.” This concerns whether a contestant on Monty Hall’s game show, *Let’s Make a Deal*, who has chosen one of three doors, should or should not switch doors after Hall has revealed that one of the doors not chosen does not hide the car. Most people intuitively declare that there’s no advantage to switching because the chances are 50/50 between the two remaining doors.

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However, in her “Ask Marilyn” column in *Parade Magazine*, vos Savant said that there is a two-thirds probability the car is behind the remaining door and a one-third probability the contestant is correct on his initial choice. The contestant should always switch.

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## Comments

## Logic "Problems"

That's the thing about logic problems. The result can often be strongly influenced by how the problem is posited or how the details are presented.

In this case, applying your Monty Hall reasoning to the Deal or No Deal quandary, if you "fenced off" the remaining 23 cases, there would be a 95.83% chance the $1 million case was in that fence, and a 4.17% chance it was in your fenced off case. Then, if 22 cases had been opened without revealing the $1 million case, you could be sure that your case still had a 4.17% chance of containing $1 million, but the other case had a 95.83% chance of containing $1 million, similar to Ms. Savant stating that the remaining door had a 2/3 chance of containing a car.

Applying your statistical demonstration to the Monty Hall problem returns an analagous result. Your door has a 1/3 chance of containing a car, and there is a 2/3 chance it does not. There are 2! paths to opening the remaining 2 doors, and 1! paths that the final door contains the car. Therefore, 1!/2! = 0.5, which we multiply by the original 2/3 to obtain a result of...1/3 ! So the final door still only has a 1/3 chance of containing the car (at least using this method of analysis).

The main difference in how these two problems are presented is that Monty knows which door has the car, and Howie's contestant does not. However, removing the randomness from the pathways doesn't change the statistics, merely the results.

## Andrew, You cannot use the

Andrew, You cannot use the Monty Hall reasoning with the "Deal Or No Deal" situation because Monty knows where the car is hidden and ALWAYS opens a door to show that there is no car behind that door. Therefore the contestant is always faced with the option to switch or not switch. In "Deal Or No Deal", if the $1 million case is left among the 23 remaining cases, it can be revealed any time during the opening process - Nobody knows where it is, plus it is the contestant, not the host, who decides what cases are opened and in what order. However there is only a 0.04347826 probability that you can choose 22 of the remaining cases and leave the $1 million case to the end. Therefore, the probability that the $1 million was not chosen (0.9583333) and left unopened to the end (0.04347826) is the product of those two porbabilities. equalling 0.416666 and there is no advantage for contestant to switch or not switch at that point.

Going back to the Monty Hall situation, suppose there were 100 doors to choose from. You choose one and then Monty (again, he knows where the car is) starts opening doors one at a time until only one is left. You would darn sure switch in that case because the chance that you selected the car intitially is just 1%!!!

## What does experience tell us.

Probabilities are a great theoretical exercise, but we cannot forget to validate our hypothesis with real data. If we analyze the Let's Make a Deal results we can get the real probabilities of changing your selection or even better helping us make a better selection to begin with.

## Indeed!

A review of the videotapes of"Let's Make a Deal" showed clearly that when contestants switched doors, they won almost exactly 2/3's of the time and only won 1/3 of the time when they did not switch. This work has been published several times in the past (see "The Drunkard's Walk - How Randomness Rules Our Lives" by Leonard Mlodinow). There have also been hundreds and hundreds of student experiments and computer simulations which confirm these results. Thanks for your comment!

## Re: What Experience Tells Us

@ WARO, Wikipedia has a couple of links on, or related to, the Monty Hall (MH) Problem. In fact, it includes published results of a Monte Carlo simulation of the MH Problem, which shows a convergence of 2/3 and 1/3 success rates, respectively.

## Oooops!

I have been told that "Deal Or No Deal" starts with 26 cases instead of the 24 noted in my article. I have not watched the program for a long time, so my memory may be a little fuzzy! Anyway, the calculations are the same. Choosing from 26 cases means the probability of choosing the $1 million is 1/26 = 0.038462, meaning also that the $1 million case remains on the upper stage with a probability of 0.961538. The probability of choosing 24 of the remaining 25 cases and leaving the $1 million case to the last is 24!/25! x 0.961538 = 1/25 x 0.961538 = 0.038462, the same probability as choosing the $1 million case at the start of the game. Still, no advantage or disadvantage to switching cases at the end.

sjm

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