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Six Sigma

Published: Wednesday, March 23, 2011 - 06:00

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.

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.

Throughout the years, there have been many articles and even books written about the Monty Hall Problem. Most analyses include Bayes Theorem and other complex mathematical approaches. I learned a few years ago from Donald J. Wheeler that the simplest analysis that gives you insight is the best analysis. So I offer a simple analysis of the Monty Hall Problem as follows.

I have had many (sometimes heated) conversations about this problem. The one way I have almost always diffused the opposition is to take the following path in logic:

**Me:** “Suppose you choose a door, any door, and build a fence around that door. What is the probability that you have selected the car?”**Opposition:** “One-third.”**Me:** “Agreed!”

**Me:** “Now let’s build a fence around the other two doors. What is the probability that the car is inside that fence?”**Opposition:** “Two-thirds.”**Me:** “Agreed!”

**Me:** “In summary, we have now agreed that there is a one-third probability that you chose the door with the car and built a fence around it, and a two-thirds probability that the car is in one of the other doors within the second fence. Correct?”**Opposition:** (after some hesitancy) “Yes.”

**Me:** “So, at this point, if Monty offered to let you take *both* the other two doors in place of the original door, would you switch?”**Opposition:** “Probably.”**Me:** “Yes, because you would double your chances of winning the car, right?”**Opposition:** “Right.”

**Me:** “However, Monty opens one of the other two doors to reveal that there is no car behind that door. This event is 100-percent ensured because Monty knows where the car is located and always opens a door with a booby prize, right?”**Opposition:** (again, with hesitancy) “Yes.”

**Me:** “So how can that event possibly change the one-third probability that you originally chose the door with the car and fenced it off? Isn’t there still a two-thirds probability that the car is inside the second fence? After all, we know there is only one car, and we know that Monty is always going to open a door within the second fence that does not reveal the car. Right?”**Opposition:** “OK… I’m convinced.”

Now fast-forward from Monty Hall and *Let’s Make a Deal* to 2005 when Howie Mandel came into our living rooms with *Deal or No Deal*. In this game, the contestant chooses one out of 24 cases having various amounts of money indicated in them from $0.01 to $1 million. The contestant hopes to choose the $1 million case and is asked to choose from among the remaining cases to reveal their contents. After revealing successive cases not previously chosen, the “Banker” offers to buy the original case the contestant chose. The contestant can “sell” his case or proceed to reveal more cases’ contents and receive more offers.

As Mandel often tells the contestant, “This is a game of luck, timing, and guts. What I want to know is do you have the guts? Deal or no deal?” This continues until the contestant sells his chosen case or until only the original case and one other are left to be opened. Mandel then offers a twist: Does the contestant want to switch cases (… shades of *Let’s Make a Deal)*? What should the contestant do to maximize his chances of having the $1 million case (assuming the $1 million case has not been revealed yet)? I now offer a simple analysis to *Deal or No Deal* as follows.

The probability that the contestant chose the $1 million case from the original 24 is given by 1/24 = 0.041666…. That means that there is a probability of 0.95833… that the $1 million case was not chosen. When the contestant chooses the first case to be revealed, she has 23 choices. Then she has 22 choices, 21 choices, and so on, until all cases will be revealed. Therefore, there are 23! paths that can be taken to open the remaining 23 cases. 23! = 23 × 22 × 21 ×… × 2 × 1. To get to the point that all but one of the remaining cases are opened, with the $1 million case being left as the last case, the contestant must reveal 22 cases, and there would be 22! ways to do this if the $1 million case had not been chosen from the original 24.

So, if the $1 million case was not chosen by the contestant at the beginning of the game, the probability of being able to choose 22 of the remaining 23 cases without revealing the $1 million case would be 22!/23! = 1/23 = 0.04347826. This probability is just slightly higher than the 0.041666… probability that the contestant chose the $1 million case to begin with, and one might be tempted to switch cases to increase one’s probability of winning the $1 million.

However, we are not quite finished yet. We still have to multiply by 0.958333… because we are calculating the probability that the case was not chosen *and* that the contestant was able to choose 22 cases without revealing the $1 million. So 0.04347826 × 0.958333… = 0.041666…, which is exactly the same as choosing the $1 million from the original 24 cases; and we can see that there would be no advantage to switching or not switching.

Many people intuitively feel that the contestant should switch because of what they have learned from Monty Hall. However, it truly makes no difference in this case. Of course if the two remaining values left were $750,000 and $1 million, you might not care; but if the two remaining values are $10 and $1 million, you may be kicking yourself for the rest of your life for switching or not switching.