Will You Choose Door #1 or Door #2? Understanding the Monty Hall Problem

Understanding the Monty Hall Problem

The television game show Let’s Make a Deal hit the airwaves in 1962. One of the most popular features of the program involved the host, Monty Hall, giving the contestant the option of choosing one of three doors. Behind one of the doors was a nice prize, such as a new car. The other two doors opened to something less than desirable, such as a goat.

Once the contestant makes a choice, Monty Hall, without letting the contestant know what is on the other side of his or her choice, opens one of the other doors, invariably revealing a goat. Then, with only one door other than the contestant’s choice remaining closed, Monty would ask if the contestant wished to stay with the chosen door or switch.

“No other statistical puzzle comes so close to fooling all the people all the time [and] even Nobel physicists systematically give the wrong answer.”

Massimo Piattelli Palmarini

What would you do? What should you do? The correct answer is more than random chance. It is rooted in statistical probability. It is also highly counter-intuitive.

Welcome to the infamous “Monty Hall Problem.”

The problem is typically presented thusly:

You are a player on a game show and are shown three identical doors. Behind one is a car, behind the other two are goats. Monty Hall, the host of the show, asks you to choose one of the doors. You do so, but you do not open your chosen door. Monty, who knows where the car is, now opens one of the doors. He chooses his door in accordance with the following rules:

1. Monty always opens a door that conceals a goat.

2. Monty never opens the door you initially chose.

3. If Monty can open more than one door without violating rules one and two, then he chooses his door randomly.

After Monty opens his door, he gives you the option of sticking with your original choice or switching to the other unopened door. What should you do to maximize your chances of winning the car?

Intuitively, a person might be drawn to one of two conclusions:

1. The probability of the player choosing correctly was one out of three and that nothing done by the host changed those odds.

2. The initial probability that the player picked the right door was one out of three, but when Monty removed one of the options, that changed things. If the player makes a choice now, the odds of getting the right door are one out of two, so it would be better to change doors.

Monty Hall (1921-2017)

To come to the correct conclusion (which is none of the above), we need to leave intuitiveness behind and look at some math.

The first part of the problem is easy. When given a choice of three doors and no other information that suggests the correct answer, the player has a 1/3 chance of choosing correctly. Let’s say that the player selects Door #1.

Here is where it starts to get a little murky, but trust us — it will all make sense by the end.

After the player makes his or her choice, Monty will also pick a door. Unlike the contestant, however, Monty knows which door conceals the new car. Armed with that knowledge, he selects Door #3 which, when opened, reveals a rather confused-looking goat.

The remaining door, Door #2, has either the car or another goat, but what is the likelihood of either possibility? Knowing the answer will help the player determine whether to take Monty up on his offer to switch doors.

Consider the problem from a different angle. When the player made his or her choice, what was the likelihood that the car was among the doors that were not chosen? Obviously, there was a 2/3 probability of that happening. When Monty opened Door #3, he did nothing to change those numbers; all he did was reveal the contents behind one of those doors. When he gives the contestant the option of switching doors, he effectively gives that player the choice, “Do you want to stick with your door, which has a 1/3 chance of being a winner, or select from the only remaining option of those doors that had a 2/3 chance of winning?”

When you look at it that way, the choice is obvious. Although the initial choice of doors only yielded a 1/3 chance of getting a new car, taking Monty up on his offer to switch doors increases the likelihood of winning to 2/3.

Still not convinced? Try looking at it this way….. Suppose there are 100 doors, instead of 3. You pick Door #1. You know that it is highly unlikely that you picked the right door, right? You have a 1% chance of nailing the winning door at that point. In all likelihood, the new car is behind a door that you didn’t choose. To be precise, there is a 99% likelihood that your new set of wheels hides behind one of the other choices.

Fortunately, Monty is going to do you a favor. Remember that he knows which door hides your hot new ride. Armed with that knowledge, he opens 98 doors — all releasing a bunch of bleating goats. Now there are only two doors that remain closed: the one you chose and Door #100.

When you chose Door #1, you didn’t really get your hopes up that you would be getting a new car, did you? You knew there was just a 1% chance that you got it right, and you knew there was a 99% chance that the car was anywhere other than behind your door. At this stage in the game, there are, effectively, two players: you and Monty Hall. You have a 1% chance of winning with your door. Monty has a 99% chance of winning with his doors. There are two big differences between the two of you: Monty gets 98 chances to open doors, and he knows which of the 100 doors is the actual winner.

As Monty goes through the process of opening 98 doors, did anything happen to change the likelihood that you picked the right one? Each door may narrow the field in terms of which of Monty’s doors does or does not have a car, but the fact that he has a 99% overall of winning, compared to your 1%, has not changed.

If you think the chances of Door #100 hiding your car are equal to those of your Door #1, you may have forgotten that Monty Hall knows which of them has the winning prize. Because of this, the 98 doors he opened were not random. There was zero probability that a car was hiding behind any of them.

Now, with only two doors remaining closed, Monty asks you if you want to stay with your choice or switch to Door #100. Basically, he is asking you if you want to change places with him. Do you want to keep your 1% likelihood of winning, or would you care to have his 99% odds?

If you were stumped by this one, you’re in good company. The question has been discussed in many venues, including Scientific American and American Statistician. It was considered by its largest audience when Marilyn vos Savant discussed it in a column in Parade magazine in 1990. It was in response to this column that nearly 1,000 readers with Ph.D. degrees wrote to her, insisting that she was wrong.

The Monty Hall problem continues to be a fertile source of discussion, not only among statisticians but also psychologists. In addition to the counter-intuitive mathematical aspects, the problem seems to trigger a certain stubborn defiance among those who insist upon the expected answer.

Cognitive psychologist Massimo Piattelli Palmarini observed, “No other statistical puzzle comes so close to fooling all the people all the time [and] even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer.”

Psychologists have proposed a number of possible reasons for this phenomenon, such as the Endowment Effect (people tend to overvalue the winning probability of whatever choice has already been made); the Status Quo Bias (people tend to prefer those decisions already made and are uncomfortable with change); and the Errors of Omission vs. Errors of Commission Effect (all other things being equal, people prefer to make errors through inaction as opposed to action).

Despite the counter-intuitiveness of the solution and the stubborn insistence of many to refuse to accept the correct answer, computer simulations and countless statistical studies confirm the fact that the contestant has a much better chance of winning by switching doors. One study published in the Journal of Comparative Psychology showed that while humans may struggle to comprehend the truth behind the Monty Hall Problem, birds do not. When presented with a similar dilemma, pigeons adjusted their behavior to conform with the statistical probability.

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