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Monty Hall problem

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The Monty Hall problem is one of several names by which the following question is known:

Assume that you are a guest in a game show. The host shows you three doors and tells you that behind one of these doors there is a car while behind the other two there are goats. He then asks you to choose a door and, after you have made your choice, he opens another door, revealing a goat. Finally, he asks you whether you want to win whatever is behind the door you have chosen first or whether you want to switch and win the prize behind the other still closed door.
Question: In order to maximize your chances to win the car, should you switch or not?

Assuming that no additional information is known and that the host always allows you to switch (independent of whether the first choice wins or not), the correct answer is:

Switching doubles the winning chances (from one-third to two-thirds).

However, almost everyone, when confronted with the question for the first time, answers incorrectly that there is no need to switch because the two closed doors are equally likely to have the car behind them, and many reject the correct answer even after it has been explained to them. Because of this conflict between intuition and probabilistic reasoning the Monty Hall problem is often called the Monty Hall paradox.

The key to accepting and understanding the solution is to realize that the (subjective) probabilities relevant for the decision are not determined by the situation (two doors closed) alone but also by what is known about the development that led to this situation.

Switching will, in the long run, win the car in two thirds of the cases, because in only one third of them the door first chosen will hide the car and rejecting the offer to switch will win it. In all the remaining cases the car is behind one of the other two doors, and switching wins because the host has already revealed which of those two is the losing one.
Or putting it differently: Assuming the (wrong) intuitive answer of equal chances would mean that, after the host has opened the door, the initial door would suddenly win in half (instead of only one third) of all cases.

The problem has a long history: It is closely related to Bertrand's box problem (1889), equivalent to a problem posed by Martin Gardner (1959) and appeared in the present form in 1975, but became famous world-wide in 1990 after it was presented in a popular weekly magazine, and it has been causing endless disputes and arguments since then. Other names for the problem are the three-doors problem, the car-and-the-goats problem (or simply the goats problem), and the quizmaster problem.

The origins of the problem

The Monty Hall problem was introduced by biostatistician Steve Selvin in a letter to the journal The American Statistician in 1975. It became famous in 1990 with its presentation in a popular weekly column called "Ask Marilyn" in Parade magazine. The column's author, Marilyn vos Savant, was, according to the Guiness Book of Records at the time, the person with the highest IQ in the world. The problem itself is named after the stage-name of an actual quizmaster, Monty Halperin (or Halparin, according to some sources), on a long-running 1960's TV show "Let's make a Deal", though the events related in the Monty Hall problem never actually took place [1], [2].

Rewriting in her own words a problem posed to her by a correspondent, Craig Whitaker, vos Savant asked the following:

Suppose you’re on a game show, and you’re given the choice of three doors: behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Almost everyone, on first hearing the problem, has the immediate and intuitive reaction that the two doors left closed, Door 1 and Door 2, must be equally likely to hide the car, and that therefore there is no point in switching. However, some thought shows that this immediate reaction must be wrong. Consider many, many repetitions of the game, and suppose (for the sake of argument) that the car is hidden each time anew completely at random behind one of the three doors, while the contestant always starts by picking Door No. 1. In the long run, the car will be behind his initially chosen door, Door 1, on one third of the repetitions of the game. If he never switches doors, he'll take the car home with him on precisely those repetitions of the show where his initial choice was actually right - that's one third of the time. On the other hand, if he always switches, he'll go home with the car two thirds of the time - he'll win the car by "switching" exactly on every occasion when he would not win it by "staying".

Note that we are assuming, as most readers do and as vos Savant later explained was her intention, that whatever choice is initially made by the contestant, the host is surely going to open a different door revealing a goat and offer the option to switch.

One could say that when the contestant initially chooses Door 1, the host is offering the contestant a choice between his initial choice Door 1, or Doors 2 and 3 together.

By changing one aspect of the problem, this way of understanding why the contestant indeed should switch may become even more compelling to the reader. Consider the 100-door problem: 99 goats and one car. The player chooses one of the 100 doors. Let's say that he chooses door number 1. The host, who knows the location of the car, one by one opens all 99 of the other doors but one - let's say that he skips door number 38. Would you switch?

The discussion till now used a frequentist picture: probability refers to relative frequency in many repetitions. Also, it didn't address the issue of whether the specific door opened by the host is relevant. Could it be that the decision to switch should depend on whether the host opens Door 2 or Door 3? These topics are taken up in the next section.

Alternative arguments

The solution in this section takes explicit account of which door was opened by the host: Door 2 or Door 3 - in order to argue that this does not change the answer. Moreover, in this solution "probability" is used in the ordinary daily-life Bayesian or subjectivist sense: that is to say, probability statements are supposed to reflect the state of knowledge of one person. To be specific, that person will be a contestant on the show who initially knows no more than the following: he'll choose a door; the quizmaster (who knows where the car is hidden) will thereupon open a different door revealing a goat and make the offer that the contestant switches to the remaining closed door. For our contestant, initially all doors are equally likely to hide the car. Moreover, if he chooses any particular door, and if the car happens to be behind that particular door, then as far as our contestant is concerned the host is equally likely to open either of the other two doors.

The contestant initially chooses door number 1. Initially, his odds that the car is behind this door are 2 to 1 against: it is two times as likely for him that his choice is wrong as that it is right.

The host opens one of the other two doors, revealing a goat. Let's suppose that for the moment, the contestant doesn't take any notice of which door was opened. Since the host is certain to open a door revealing a goat whether or not the car is behind Door 1, the information that an unspecified door is opened revealing a goat does not change the contestant's odds that the car is indeed behind Door 1; they are still 2 to 1 against.

Now here comes the further detail which we will take account of in this solution: the contestant also gets informed which specific door was opened by the host - let's say it was Door 3. Does this piece of information influence his odds that the car is behind Door 1? No: from the contestant's point of view, the chance that the car is behind Door 1 can't depend on whether the host opens Door 2 or Door 3 - the door numbers are arbitrary, exchangeable.

Therefore, also knowing that the host opened specifically Door 3 to reveal a goat, the contestant's odds on the car being behind his initially chosen Door 1 still remain 2 to 1 against. He had better switch to Door 2.

Explicit computations

Students of probability theory might feel uneasy about the informality (the intuitive nature) of the last step. Ordinary people's intuition about probability is well known to be often wrong — after all, it is ordinary intuition which makes most people believe there is no point in switching doors! To feel more secure, students of probability theory might consider the mathematical concept of symmetry and use the law of total probability to show how symmetry leads to statistical independence between the events "Car is behind Door 1" and "Host opens Door 3", when it is given that the contestant chose Door 1. Alternatively, they might like to explicitly use Bayes' rule: posterior odds equals prior odds times likelihood ratio. They just have to check that under the two competing hypotheses (whether or not the car is behind the door chosen by the contestant, Door 1), the fact that it is Door 3 (rather than Door 2) which gets opened by the host has the same probability 1/2.

For some readers, numbers speak louder than words. The following table should be self-explanatory.

  Door opened by host (Door 1 chosen by contestant)
Initial arrangement
(probability)
Open D1
(probabilty)
Open D2
(probability)
Open D3
(probability)
Joint
probability
Win by
staying
Win by
switching
Car Goat Goat
(1/3)
No Yes
(1/2)
No 1/3 x 1/2 Yes
(1/6)
No
No No Yes
(1/2)
1/3 x 1/2 Yes
(1/6)
No
Goat Car Goat
(1/3)
No No Yes
(1)
1/3 x 1 No Yes
(1/3)
Goat Goat Car
(1/3)
No Yes
(1)
No 1/3 x 1 No Yes
(1/3)
Note that the host has limited choices when the contestant chooses incorrectly

We observe that the player who switches wins the car 2/3 of the time. We also see that Door 3 is opened by the host 1/2 = 1/6+1/3 of the time (row 2 plus row 3), as must also be the case by the symmetry of the problem with regard to the door numbers — either Door 2 or Door 3 must be opened and the chance of each must be the same, by symmetry.

Winning by switching in combination with Door 3 being opened occurs 1/3 of the time (row 3). The conditional probability of winning by switching, given Door 3 is opened, is therefore (1/3)/(1/2)=2/3. Since this is the same as the overall chance 2/3 of winning by switching, we see that knowing the identity of the opened door doesn't change the chance of winning by switching. Not only does the switcher win 2/3 of the time, he also wins 2/3 of the time when Door 3 is opened by the host, and 2/3 of the time when Door 2 is opened by the host.

In other words, the combined chance of winning by switching and Door 3 (rather than Door 2) being opened, 1/3, equals the product of the separate chances of "the car being behind the other door", 2/3, and "host opens Door 3", 1/2. Whether or not the car is behind the door not opened by the host is statistically independent of whether the host opens Door 2 or Door 3.

This last fact could have predicted in advance, by the symmetry of the problem. The contestant may simply ignore the door numbers: they do not change his chances of winning by staying or by switching.

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