Monty Hall problem

The Monty Hall problem is a puzzle involving probability, loosely based on the American game show Let's Make a Deal. The name comes from that of the show's host, Monty Hall. The problem is also called the Monty Hall paradox; it is a veridical paradox in the sense that the solution is counterintuitive. For example, when the problem and correct solution offered by Marilyn vos Savant appeared in her Ask Marilyn column in Parade Magazine approximately 10,000 readers, including several hundred mathematics professors, wrote to tell her she was wrong. Some of the controversy was because the Parade magazine statement of the problem is technically ambiguous since it fails to fully specify the host's behavior. However, despite completely unambiguous problem statements, explanations, simulations, and formal mathematical proofs, the correct answer is met with disbelief by many people.

Problem and solution

Problem

In search of a new car, the player picks Door 1. The game host then opens Door 3 to reveal a goat and offers to let the player pick Door 2 instead of Door 1.

A widely known statement of the Monty Hall problem appeared in a letter to Marilyn vos Savant's Ask Marilyn column in Parade Magazine (vos Savant 1990):

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?

While this is a common presentation of the problem, it leaves critical aspects of the host's behavior unstated making the problem mathematically ambiguous. In a more precise statement of the problem (Mueser and Granberg 1999) the host is constrained to always open a door revealing a goat and to always make the offer to switch:

A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. "First you point toward a door," he says. "Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door."

You begin by pointing to door number 1. The host shows you that door number 3 has a goat. Do the player's chances of getting the car increase by switching to Door 2?

The problem as generally intended also assumes that the particular door the host opens conveys no special information about whether the player's initial choice is correct. The simplest way to make this explicit is to add a constraint that the host will open one of the remaining two doors randomly if the player initially picked the car.

Solution

File:MontyHallProblemMadeEasy.jpg

There is no paradox, really.

Once the host has opened a door, the car must be behind one of the two remaining doors. Since there is no way for the player to know which of these doors is the winning door, many people assume that each door has an equal probability and conclude that switching does not matter. This "equal probability" assumption, whilst being intuitively seductive, is actually incorrect.

The player has a one in three chance of initially choosing the car and a two in three chance of initially choosing a goat. In both cases the host must reveal a goat. If the player initially chose the car, then switching loses; but if the player initially chose a goat, the host must reveal the other goat and switching wins since the only remaining door must be the door with the car.

When the player is asked whether to switch, there are three possible situations corresponding to the player's initial choice, each with probability ⅓:

• The player originally picked the door hiding goat number 1. The game host has shown the other goat.
• The player originally picked the door hiding goat number 2. The game host has shown the other goat.
• The player originally picked the door hiding the car. The game host has shown either of the two goats.

If the player chooses to switch, the player wins the car in the first two cases. A player choosing to stay with the initial choice wins in only the third case. Since in two out of three equally likely cases switching wins, the probability of winning by switching is ⅔. In other words, players who switch will win the car on average two times out of three.

The solution would be different if the host did not know what was behind each door, or if the host sometimes had the option of not offering the player the chance to switch. Some statements of the problem, notably the one in Parade Magazine, do not explicitly exclude these possibilities. For example, if the game host only offers the opportunity to switch if the contestant originally chooses the car, the probability of winning by switching is 0%. In the problem as stated by Mueser and Granberg, it is because the host must reveal a goat and must make the offer to switch that the player has a ⅔ chance of winning by switching.

Aids to understanding

The most common objection to the solution is that the past can be ignored when assessing the probability; thus ignoring the first choice of door, and the host's choice of which door to open. If all the player knew was that there was one door that contained a goat, there would indeed be a 50–50 choice between the other two; and many people leap to the conclusion that this is also true in this problem.

Although ignoring the past works fine for some games, like coin flipping, it does not work for all games. In this case what should be ignored is the opening of the door. The player actually chooses between the originally picked door and the other two — opening one is simply a distraction. There is only one car and it does not move. The original choice divides the possible locations of the car between the one door the player picks with a ⅓ chance and the other two with a ⅔ chance. It is already known that at least one of the two unpicked doors contains a goat. Revealing the goat therefore gives the player no additional information about the originally chosen door; it does not change the ⅔ probability that the car is still in the block of two doors.

Another possible reason for confusion is that the problem is often stated as though the host takes the player by surprise by opening the door and offering the choice. This tends to give the impression that the host is trying to confuse a player who has chosen correctly, and would mean the player did not know the rules in advance. If the player did not know the rules, that would not alter the probability in the particular case, but it would mean that the player could not definitively make the optimal choice. This confusion is dealt with in the unambiguous statement of the problem where the host explicitly relates the rules to the contestant in advance.

One reason the Monty Hall problem may be so counterintuitive is that the host is expected to be deceitful (Mueser and Granberg 1999). If the host opens a door only when the player has chosen correctly, then when the host opens a door the player should never opt to switch.

Increasing the number of doors

It may be easier to appreciate the result by considering a 100 doors instead of just three. In this case there are 99 doors with goats behind them and one door with a prize. The player picks a door. The game host then opens 98 of the other doors revealing 98 goats — imagine the host starting with the first door and going down a line of 100 doors, opening each one but skipping over only the player's door and one other door. The host then offers the player the chance to switch to the only other unopened door. On average, in 99 out of 100 times the other door will contain the prize, as 99 out of 100 times the player first picked a door with a goat. A rational player should switch.

There is a reasonable question to this logic: if the number of doors is increased, why does this explanation assume the host would open 98 doors to make the problem similar to the original? Why does the host not open 33 doors instead? In the 3 door game the player can stay with the original choice or switch to one other door, so the player in the 100 door game should also be presented with the option of staying or switching to one other door. The three-door game is misleading because the player is always presented with ⅓ proportions. There is a ⅓ chance of winning, the host reveals ⅓ of the mystery, and the player is allowed to switch to the other ⅓ option. All options seem equal. This is an essential ingredient for the counter-intuitiveness of the original problem.

Even if only one of the 100 doors is opened, switching still increases the player's chances of finding a car. The ⁹⁹⁄₁₀₀ chance that the car is not behind the door the player picked is spread evenly over 98 doors after the host reveals one goat. Each of those 98 doors, that is all doors other than the one the player picked and the one the host reveals, has a ⁹⁹⁄₉₈₀₀ chance of having the car, so by switching the player slightly improves the chances of winning the car — from a 0.0100 chance to just over 0.0101. This is an improvement of ⁹⁹⁄₉₈ or just over 1%.

The same algorithm can be followed for any number of doors, N. The algorithm is "Choose a door, eliminate remaining losing doors until 2 doors remain, decide to switch or not." This algorithm can be followed for N = 3 or N = 100. The higher N values demonstrate the same mathematical principle in a more obvious way.

Venn diagram

The probability that the car is behind the remaining door can be calculated using Venn diagrams. After choosing Door 1, for example, the player has a ⅓ chance of having selected the door with the car, leaving a ⅔ chance between the other two doors, as shown below. Note that there is a 100% chance of finding a goat behind at least one of the two unchosen doors because there is only one car.

The host now opens Door 3. Since the host knows what is behind the doors and must always open a door revealing a goat, opening this door does not affect the chance that the car is behind the originally chosen door which remains ⅓. The car is not behind Door 3, so the entire ⅔ probability of the two unchosen doors is now carried only by Door 2, as shown below. Another way to state this is that if the car is behind either door 2 or 3, by opening Door 3 the host has revealed it must be behind Door 2.

Decision tree

Tree listing the probability of every possible outcome

More formally, the scenario can be depicted in a decision tree.

In the first two cases, wherein the player has first chosen a goat, switching will yield the car. In the third and fourth cases, since the player has chosen the car initially, a switch will lead to a goat.

The total probability that switching wins is equal to the sum of the first two events, ¹⁄₃ + ¹⁄₃ = ²⁄₃. Likewise, the probability that staying wins is ¹⁄₆ + ¹⁄₆ = ¹⁄₃.

Combining doors

Instead of one door being opened and shown to be a losing door, an equivalent action is to combine the two unchosen doors into one since the player cannot, and will not, choose the opened door. The player therefore has the choice of either sticking with the original choice of door with a ⅓ chance of winning the car, or choosing the sum of the contents of the two other doors with a ⅔ chance. The game assumptions play a role here — switching is equivalent to taking the combined contents because the game host is required to open a door with a goat.

Bayes' theorem

An analysis of the problem using the formalism of Bayesian probability theory (Gill 2002), makes explicit the role of the assumptions underlying the problem. In Bayesian terms, probabilities are associated to propositions, and express a degree of belief in their truth, subject to whatever background information happens to be known. For this problem the background is the set of game rules, and the propositions of interest are:

$C_{i}\,$ : The car is behind Door i, for i equal to 1, 2 or 3.

$H_{ij}\,$ : The host opens Door j after the player has picked Door i, for i and j equal to 1, 2 or 3.

For example, $C_{1}\,$ denotes the proposition the car is behind Door 1, and $H_{12}\,$ denotes the proposition the host opens Door 2 after the player has picked Door 1. Indicating the background information with $I\,$, the assumptions are formally stated as follows.

First, the car can be behind any door, and all doors are a priori equally likely to hide the car. In this context a priori means before the game is played, or before seeing the goat. Hence, the prior probability of a proposition $C_{i}\,$ is:

$P(C_{i}|I)\,=1/3\$.

Second, the host will always open a door that has no car behind, chosen among the two not picked by the player. If two such doors are available, each one is equally likely to be opened. This rule determines the conditional probability of a proposition $H_{ij}\,$ subject to where the car is, i.e. conditioned on a proposition $C_{k}\,$. Specifically, it is:

$P(H_{ij}|C_{k},\,I)\,\,=\,{\begin{cases}\,\\\,\\\,\\\,\end{cases}}$	$\,0\,$		if i = j, (the host cannot open the door picked by the player)
	$\,0\,$		if j = k, (the host cannot open a door with a car behind it)
	$\,1/2\,$		if i = k, (the two doors with no car are equally likely to be opened)
	$\,1\,$		if i ≠k and j ≠ k, (there is only one door available to open)

The problem can now be solved by scoring each strategy with its associated posterior probability of winning, that is with its probability subject to the host's opening of one of the doors. Without loss of generality assume, by re-numbering the doors if necessary, that the player picks Door 1, and the host then opens Door 3, showing him or her a goat. In other words, the host makes proposition $H_{13}\,$ true.

The posterior probability of winning by not switching doors, subject to the game rules and $H_{13}\,$, is then $P(C_{1}|H_{13},\,I)$. Using Bayes' theorem this is expressed as:

$P(C_{1}|H_{13},\,I)={\frac {P(H_{13}|C_{1},\,I)\,P(C_{1}|I)}{P(H_{13}|I)}}$ .

By the assumptions stated above, the numerator of the right-hand side is:

$P(H_{13}|C_{1},\,I)\,P(C_{1}|I)=1/2\times 1/3=1/6$.

The normalizing constant at the denominator can be evaluated by expanding it using the definitions of marginal probability and conditional probability:

${\begin{array}{lcl}P(H_{13}|I)&{}=&P(H_{13},\,C_{1}|I)+P(H_{13},\,C_{2}|I)+P(H_{13},\,C_{3}|I)\\&{}=&P(H_{13}|C_{1},\,I)\,P(C_{1}|I)\,+\\&&P(H_{13}|C_{2},\,I)\,P(C_{2}|I)\,+\\&&P(H_{13}|C_{3},\,I)\,P(C_{3}|I)\\&{}=&1/2\times 1/3+1\times 1/3+0\times 1/3\\&{}=&1/2\end{array}}$

Putting the two together yields:

$P(C_{1}|H_{13},\,I)=1/6\,/\,1/2=1/3$.

The probability of winning by switching the selection to Door 2, $P(C_{2}|H_{13},\,I)$, can now be evaluated by requiring that the posterior probabilities of all the $C_{i}\,$ propositions add to 1. That is:

$1=P(C_{1}|H_{13},\,I)+P(C_{2}|H_{13},\,I)+P(C_{3}|H_{13},\,I)$.

There is no car behind Door 3 since the host opened it, so the last term must be zero. This can be proven using Bayes' theorem and the previous results:

${\begin{aligned}P(C_{3}|H_{13},\,I)&={\frac {P(H_{13}|C_{3},\,I)\,P(C_{3}|I)}{P(H_{13}|I)}}\\&={\frac {0\times 1/3}{1/2}}=0\end{aligned}}$

Hence:

$P(C_{2}|H_{13},\,I)=1-1/3-0=2/3$

which shows that the winning strategy is to switch doors.

Simulation

A simple way to demonstrate that a switching strategy really does win two out of three times on the average is to simulate the game with playing cards. Three cards from an ordinary deck are used to represent the three doors; one 'special' card such as the Ace of Spades should represent the door with the car, and ordinary cards, such as the two red deuces, represent the goat doors.

The simulation, using the following procedure, can be repeated several times to simulate multiple rounds of the game. One card is dealt at random to the 'player', to represent the door the player picks initially. Then, looking at the remaining two cards at least one of which must be a red two, the 'host' discards a red two. If the card remaining in the host's hand is the Ace of Spades, this is recorded as a round where the player would have won by switching; if the host is holding a red two, the round is recorded as one where staying would have won.

By the law of large numbers, this experiment is likely to approximate the probability of winning, and running the experiment over enough rounds should not only verify that the player does win by switching two times out of three, but show why. Two times out of three, after one card has been dealt to the player, the Ace of Spades is in the hosts's hand. At that point, it is already determined whether staying or switching will win the round for the player.

If this is not convincing, the simulation can be done with the entire deck, dealing one card to the player and keeping the other 51. In this variant the Ace of Spades goes to the host 51 times out of 52, and stays with the host no matter how many non-Ace cards are discarded.

Other host behaviors

Possible host behaviors in unspecified problem
Host behavior	Result
The host, to minimize the show's budget for prizes, only offers the option to switch when the player's initial choice is the winning door.	Switching always yields a goat.
The host, to maximize contestants' winnings, only offers the option to switch when the player has chosen incorrectly.	Switching is always successful.
The host does not know what lies behind the doors.	The game ends prematurely a third of the time. If the prize is still hidden, switching is successful half of the time.
The host has the option of opening the door that the contestant picked.	If there is no car behind the contestant's door, then switching to one of the other doors gives a 50% chance of winning.
The host acts as noted in the specific version of the problem.	Switching is successful two-thirds of the time.

In some versions of the Monty Hall problem the host's behavior is not fully specified. For example, the version published in Parade in 1990 did not specifically state that the host would always open another door, or always offer a choice to switch, or even never open the door revealing the car. Without specifying these rules, the player does not have enough information to conclude switching is the best option. The table shows possible host behaviors and the impact on the success of switching.

Variants

Two players

In this variant two players are each allowed to choose different door. The game host eliminates the player or a random one who has chosen a door hiding a goat, opens the eliminated player's door, and then offers the remaining player a chance to switch. Should the remaining player switch?

The answer is no. Switching in this game wins if and only if both players originally picked goats; the likelihood of this is only ⅓. By sticking with the original choice the remaining player wins in the remaining ⅔ of the cases. So stickers will win twice as often as switchers.

There are three possible scenarios, all with probability ⅓:

• Player 1 picks the door with the car. The host must eliminate player 2. Switching loses.
• Player 2 picks the door with the car. The host must eliminate player 1. Switching loses.
• Neither player picks the car. The host eliminates one of the players randomly. Switching wins.

Player 1 is the remaining player in the first case and half the time in the third case. Switching loses twice as often as it wins: ⅓ chance of being the remaining player and switching losing vs. ¹⁄₆ chance of remaining and switching winning. Similarly, player 2 is the remaining player in the second case and half the time in the third, and also loses twice as often by switching. Regardless of which player remains, this player has a ⅔ probability of winning with the sticking strategy.

The two player game, from the final player's point of view, resembles the single player game: the player chooses a door, a goat is revealed behind another door, and the player is given the opportunity to switch. However, the significant difference is that one player is eliminated. The process of surviving the elimination improves the remaining player's chances of having chosen the car from ⅓ to ⅔. Another way to look at this is that the chances of the remaining player having not chosen the car initially is a combined probability: it is the chance of not choosing the car initially and not being the eliminated player: ⅔ ×½ = ⅓. The only other scenario for the remaining player is choosing the car and since the two possible outcomes must have a probability of 1 the probability of having the car is now ⅔.

The two player game is exactly the same as the one player game, except in reverse. In the one player game, it is the player's chosen door that is guaranteed to not be opened, and which therefore retains the original probability of ⅓. In the two person game, it is the unchosen door that is guaranteed to not be opened, and which therefore retains the original probability. If there were a three person game, in which one of the goat doors is randomly chosen, then no door can be categorized as guaranteed not to be opened, and therefore none of them retain the original probability of ⅓. In such a game, there is a true symmetry between the doors, and there would be no benefit to either switching or not switching.

Sequential doors

There is a generalization of the original problem to n doors. In the first step, the player chooses a door. The game host then opens some other door that is a loser. If desired, the player may then switch to another door. The game host then opens another as-yet-unopened losing door, different from the player's current choice. Then the player may switch again, and so on. This continues until there are only two unopened doors left: the player's current choice and another one. How many times should the player switch, and when, if at all?

The best strategy is: stick with the first choice all the way through but then switch at the very end. With this strategy, the probability of winning is (n−1)/n (Bapeswara Rao and Rao 1992).

This problem appears similar to the television show Deal or No Deal, which typically begins with 26 boxes. The player selects one to keep, and then randomly picks boxes to open from amongst the rest. In this game, even until the end, the box the player initially selects and all boxes left unrevealed are equally likely to be the winner. The distinction is that any box the player picks to open might reveal the grand prize, thereby eliminating it from contention. Monty on the other hand, knows the contents and is forbidden from revealing the winner. Because the Deal or No Deal player is just as likely to open the winning box as a losing one, the Monty Hall advantage is lost. Assuming the grand prize is still left with two boxes remaining, the player has a 50/50 chance that the initially selected box contains the grand prize.

Quantum version

A quantum version of the paradox illustrates some points about the relation between classical or non-quantum information and quantum information, as encoded in the states of quantum mechanical systems. The three doors are replaced by a quantum system allowing three alternatives; opening a door and looking behind it is translated as making a particular measurement. The rules can be stated in this language, and once again the choice for the player is to stick with the initial choice, or change to another "orthogonal" option. The latter strategy turns out to double the chances, just as in the classical case. However, if the show host has not randomized the position of the prize in a fully quantum mechanical way, the player can do even better, and can sometimes even win the prize with certainty (D'Ariano et al 2002).

Forgetful host

In November 2006 the problem returned to vos Savant's column in a slightly different form. In the variation, Monty Hall forgets which door hides the car. He opens one of the doors at random and is relieved when a goat is revealed. Asked whether the contestant should switch, vos Savant correctly replied "If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch" (vos Savant, 2006).

In this variation, at the point the host opens the door, there are six possible situations corresponding to the player's initial choice, each with probability ¹⁄₆:

• The player originally picked the door hiding goat number 1. The game host shows the other goat.
• The player originally picked the door hiding goat number 1. The game host shows the car.
• The player originally picked the door hiding goat number 2. The game host shows the other goat.
• The player originally picked the door hiding goat number 2. The game host shows the car.
• The player originally picked the door hiding the car. The game host shows goat 1.
• The player originally picked the door hiding the car. The game host shows goat 2.

Since the problem statement says the host does not show the car, only the four cases where the host shows a goat need to be considered. If the player chooses to switch, the car is won in two of these cases. A player choosing to stay with the initial choice also wins in two cases. Since in two out of four equally likely cases switching wins, the probability of winning by switching in this variation is ¹⁄₂.

Similar problems

• Three cards problem
• Boy or Girl
• Three Prisoners Problem

Despite similarity in their names, the game used in the Monty Hall problem is not related to three-card Monte.

History of the problem

An essentially identical problem appeared as the Three Prisoners Problem in Martin Gardner's Mathematical Games column in Scientific American in 1959 (Gardner 1959). Gardner's version makes the selection procedure explicit, avoiding the unstated assumptions in the Parade Magazine version. This puzzle in probability theory involves three prisoners, a random one of whom has been secretly chosen to be executed in the morning. The first prisoner begs the guard to tell him which of the other two will go free, arguing that this reveals no information about whether the prisoner will be the victim; the guard responds by claiming that if the prisoner knows that a specific one of the other two prisoners will go free it will raise the first prisoner's subjective chance of being executed from ⅓ to ½. The question is whether the analysis of the prisoner or the guard is correct. In the version given by Martin Gardner, the guard then performs a particular randomizing procedure for selecting which name to give the prisoner; this gives the equivalent of the Monty Hall problem without the usual ambiguities in its presentation.

In 1975, Steve Selvin wrote a pair of letters to the American Statistician (Selvin 1975a, Selvin 1975b) regarding the Monty Hall problem. The first letter presented the problem in a version close to its most popular form; the version presented in Parade Magazine 15 years later is a restatement of Selvin's version. The second letter appears to be the first use of the term "Monty Hall problem". The problem is actually an extrapolation from the game show; Monty Hall did open a wrong door to build excitement, but did not allow players to change their choice. As Monty Hall wrote to Selvin (Hall 1975):

And if you ever get on my show, the rules hold fast for you — no trading boxes after the selection.

Phillip Martin's article in a 1989 issue of Bridge Today magazine entitled "The Monty Hall Trap" (Martin 1989) presented Selvin's problem, with the correct solution, as an example of how one can fall into the trap of treating non-random information as if it were random. Martin then gives examples in the game of bridge where players commonly miscalculate the odds by falling into the same trap. Given the controversy that would arise over this problem a year later, Martin showed a remarkable lack of prescience when he stated, "Here [in the Monty Hall problem] the trap is easy to spot. But the trap can crop up more subtly in a bridge setting."

A restated version of Selvin's problem statement appeared in Marilyn vos Savant's Ask Marilyn question-and-answer column of Parade magazine in September 1990 (vos Savant 1990). Though vos Savant gave the correct answer that switching would win two-thirds of the time, vos Savant estimates 10,000 readers including several hundred mathematics professors wrote in to declare that her solution was wrong. As a result of the publicity the problem earned the alternative name Marilyn and the Goats.

In November 1990, an equally contentious discussion of vos Savant's article took place in Cecil Adams's column The Straight Dope (Adams 1990). Adams initially answered, incorrectly, that the chances for the two remaining doors must each be one in two. After a reader wrote in to correct the mathematics of Adams' analysis, Adams agreed that mathematically, he had been wrong, but said that the Parade version left critical constraints unstated, and without those constraints, the chances of winning by switching were not necessarily ⅔. Numerous readers, however, wrote in to claim that Adams had been "right the first time" and that the correct chances were one in two.

The Parade column and its response received considerable attention in the press, including a front page story in the New York Times (Tierney 1991) in which Monty Hall himself was interviewed. He appeared to understand the problem quite well, giving the reporter a demo with car keys and explaining how actual game play on Let's Make a Deal differed from the rules of the puzzle.

Over 40 papers have been published about this problem in academic journals and the popular press (Mueser & Granberg 1999).

Outside of acedemia, the problem continues to resurface. The syndicated NPR program, Car Talk, featured it as one of their weekly "Puzzlers," and the answer they featured was quite clearly explained as the correct one (Magliozzi & Magliozzi, 1998). The problem is discussed, from the perspective of a boy with Asperger syndrome, in The Curious Incident of the Dog in the Night-time, a 2003 novel by Mark Haddon. An account of mathematician Paul Erdős's first encounter of the problem can be found in The Man Who Loved Only Numbers — like many others, he initially got it wrong. The problem is also addressed in a lecture by the character Charlie Eppes in an episode of the CBS drama NUMB3RS (Episode 1.13).

See also

• Two envelopes problem
• Bertrand's box paradox
• Balls and vase problem

References

• Adams, Cecil (1990)."On 'Let's Make a Deal,' you pick Door #1. Monty opens Door #2—no prize. Do you stay with Door #1 or switch to #3?", The Straight Dope, (November 2 1990). Retrieved July 25, 2005.
• Bapeswara Rao, V. V. and Rao, M. Bhaskara (1992). "A three-door game show and some of its variants". The Mathematical Scientist 17, no. 2, pp. 89–94
• Bohl, Alan H.; Liberatore, Matthew J.; and Nydick, Robert L. (1995). "A Tale of Two Goats … and a Car, or The Importance of Assumptions in Problem Solutions". Journal of Recreational Mathematics 1995, pp. 1–9.
• D'Ariano, G.M et al (2002). "The Quantum Monty Hall Problem" (PDF). Los Alamos National Laboratory, (February 21, 2002). Retrieved January 15, 2007.
• Gardner, Martin (1959). "Mathematical Games" column, Scientific American, October 1959, pp. 180–182. Reprinted in The Second Scientific American Book of Mathematical Puzzles and Diversions.
• Gill, Jeff (2002). Bayesian Methods, pp. 8-10. CRC Press. ISBN 1-5848-8288-3.
• Hall, Monty (1975). The Monty Hall Problem. LetsMakeADeal.com. Includes May 12, 1975 letter to Steve Selvin. Retrieved January 15, 2007.
• Magliozzi, Tom; Magliozzi, Ray (1998). Haircut in Horse Town: & Other Great Car Talk Puzzlers. Diane Pub Co. ISBN 0-7567-6423-8.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Martin, Phillip (1989). "The Monty Hall Trap", Bridge Today, May–June 1989. Reprinted in Granovetter, Pamela and Matthew, ed. (1993), For Experts Only, Granovetter Books.
• Morone, A., and A. Fiore (2006), "Monty Hall's Three Doors for Dummies", No 12, series from Dipartimento di Scienze Economiche - Università di Bari.
• Mueser, Peter R. and Granberg, Donald (May 1999). "The Monty Hall Dilemma Revisited: Understanding the Interaction of Problem Definition and Decision Making", University of Missouri Working Paper 99-06. Retrieved July 5, 2005.
• Nahin, Paul J. (2000). Duelling idiots and other probability puzzlers. Princeton University Press, Princeton, NJ, pp. 192–193. ISBN 0-691-00979-1.
• Selvin, Steve (1975a). "A problem in probability" (letter to the editor). American Statistician 29(1):67 (February 1975).
• Selvin, Steve (1975b). "On the Monty Hall problem" (letter to the editor). American Statistician 29(3):134 (August 1975).
• Tierney, John (1991). "Behind Monty Hall's Doors: Puzzle, Debate and Answer?", The New York Times (21 July 1991), Sunday, Section 1; Part 1; Page 1; Column 5
• vos Savant, Marilyn (1990). "Ask Marilyn" column, Parade Magazine p. 12 (17 February 1990). [cited in Bohl et al., 1995]
• vos Savant, Marilyn (2006). "Ask Marilyn" column, Parade Magazine p. 6 (26 November 2006)

External links

• Weisstein, Eric W. "Monty Hall Problem". MathWorld.
• The Monty Hall Problem at letsmakeadeal.com (quotes Monty's letter to Steve Selvin in full)
• Monty Hall Paradox (let's make a deal) (lengthy bibliography)
• Graphical Proof of the Monty Hall Problem (tea cups and diamonds)
• Monty Hall Dilemma at cut-the-knot
• The Monty Hall Problem - A game simulation (illustration by a demonstrative, user-friendly and versatile game simulation)
• A Monty Hall Simulator in Javascript (A Simulator that you can run in your web browser. )
• Grand Illusions Explanation and various simulators
• Secret of Monty Hall (explanation from the bad-door side)
• Monty Hall Simulator (Simulators for the keep and change strategy in Perl, with data files representing the results of playing the game one million times with each strategy)
• Realtime global simulation A simulation which tallies up the results from every user who has ever played.
• The Monty Hall Problem Web Page A simple presentation of the problem and its solution.
• A tree-diagram of the Monty Hall problem under the Marilyn vos Savant assumptions
• The Game Show Problem The original question and responses on Marilyn's web site.