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Six Sigma

Published: Wednesday, March 23, 2011 - 05:00

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.

## Comments

## 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

## 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.

## 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.

## 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!

## 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%!!!