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[[File:Monty open door.svg|thumb|In search of a new car, the player picks a door, say 1. The game host then opens one of the other doors, say 3, to reveal a goat and offers to let the player pick door 2 instead of door 1.]]
The '''Monty Hall problem''' is a brain teaser, in the form of a [[probability]] puzzle ([[#refKraussandWang2003|Gruber, Krauss and others]]), loosely based on the American television game show ''[[Let's Make a Deal]]'' and named after its original host, [[Monty Hall]]. The problem was originally posed in a letter by [[Steve Selvin]] to the ''[[The American Statistician|American Statistician]]'' in 1975 {{Harv|Selvin|1975a}}, {{Harv|Selvin|1975b}}. It became famous as a question from a reader's letter quoted in [[Marilyn vos Savant]]'s "Ask Marilyn" column in ''[[Parade (magazine)|Parade]]'' magazine in 1990 {{Harv|vos Savant|1990a}}:
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The behavior of the host is key to the 2/3 solution. Ambiguities in the "Parade" version do not explicitly define the protocol of the host. However Marilyn vos Savant's ({{Harvnb|vos Savant|1990a}}) solution printed alongside Whitaker's question implies and both {{Harvtxt|Selvin|1975a}} and {{Harvtxt|vos Savant|1991a}} explicitly define the role of the host as follows
# the host must always open a door that was not picked by the contestant ([[#refMueserandGranberg1999|Mueser and Granberg 1999]]),
# the host must always open a door to reveal a goat and never the car ([[#refAdams1990|Adams 1990]])
# the host must always offer the chance to switch between the originally chosen door and the remaining closed door
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As [[Keith Devlin]] says ([[#refDevlin2003|Devlin 2003]]), "By opening his door, Monty is saying to the contestant 'There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3. I'll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner. I have not changed that. But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3.{{' "}}
Vos Savant suggests that the solution will be more intuitive with 1,000,000 doors rather than 3. {{Harv|vos Savant|1990a}} In this case there are 999,999 doors with goats behind them and one door with a prize. After the player picks a door the host opens all but 1 of the remaining doors. On average, in 999,999 out of 1,000,000 times that remaining door will contain the prize. Intuitively, the player should ask how likely is it, that given a million doors,
===Solutions using conditional probability===
====Refining the simple solution====▼
The [[Monty Hall Problem#Simple Solutions|simple solutions]] above show that a player with a strategy of switching wins the car with overall probability 2/3, i.e., without taking account of which door was opened by the host ([[#refGrinsteadandSnell2006|Grinstead and Snell 2006:137–138]] [[#refCarlton2005|Carlton 2005]]). In contrast most sources in the field of [[probability]] calculate the [[conditional probabilities]] that the car is behind door 1 and door 2 are 1/3 and 2/3 given the contestant initially picks door 1 and the host opens door 3 ({{Harvtxt|Selvin|1975b}}, [[#refMorganetal1991|Morgan et al. 1991]], [[#refChun1991|Chun 1991]], [[#refGillman1992|Gillman 1992]], [[#refCarlton2005|Carlton 2005]], [[#refGrinsteadandSnell2006|Grinstead and Snell 2006:137–138]], [[#refLucasetal2009|Lucas et al. 2009]]). The solutions in this section consider just those cases in which the player picked door 1 and the host opened door 3.▼
<!---===Criticism of the simple solutions=== Any missing stuff to be re-incorporated▼
If we assume the host opens a door at random, when given a choice, then which door the host opens gives us no information at all as to whether or not the car is behind door 1. In the simple solutions, we already observed that the probability that the car is behind door 1, the door initially chosen by the player, is initially 1/3. Moreover, the host is certainly going to open ''a'' (different) door, so opening ''a'' door (''which'' door unspecified) does not change this. 1/3 must be the average probability that the car is behind door 1 given the host picked door 2 and given the host picked door 3 because these are the only two possibilities. But these two probabilities are the same. Therefore they are both equal to 1/3 ([[#refMorganetal1991|Morgan et al. 1991]]). This shows that the chance that the car is behind door 1 given that the player initially chose this door and given that the host opened door 3 is 1/3, and it follows that the chance that the car is behind door 2 given the player initially chose door 1 and the host opened door 3 is 2/3. The analysis also shows that the overall success rate of 2/3, achieved by ''always switching'', cannot be improved, and underlines what already may well have been intuitively obvious: the choice facing the player is that between the door initially chosen, and the other door left closed by the host, the specific numbers on these doors are irrelevant.▼
As already remarked, most sources in the field of [[probability]], including many introductory probability textbooks, solve the problem by showing the [[conditional probabilities]] the car is behind door 1 and door 2 are 1/3 and 2/3 (not 1/2 and 1/2) given the contestant initially picks door 1 and the host opens door 3; various ways to derive and understand this result were given in the previous subsections.▼
Among these sources are several that explicitly criticize the popularly presented "simple" solutions, saying these solutions are "correct but ... shaky" ([[#refRosenthal2005a|Rosenthal 2005a]]), or do not "address the problem posed" ([[#refGillman1992|Gillman 1992]]), or are "incomplete" ([[#refLucasetal2009|Lucas et al. 2009]]), or are "unconvincing and misleading" ([[#refEisenhauer2001|Eisenhauer 2001]]) or are (most bluntly) "false" ([[#refMorganetal1991|Morgan et al. 1991]]).▼
Some say that these solutions answer a slightly different question – one phrasing is "you have to announce ''before a door has been opened'' whether you plan to switch" ([[#refGillman1992|Gillman 1992]], emphasis in the original).▼
The simple solutions show in various ways that a contestant who is determined to switch will win the car with probability 2/3, and hence that switching is the winning strategy, if the player has to choose in advance between "always switching", and "always staying". However, the probability of winning by ''always'' switching is a logically distinct concept from the probability of winning by switching ''given the player has picked door 1 and the host has opened door 3''. As one source says, "the distinction between [these questions] seems to confound many" ([[#refMorganetal1991|Morgan et al. 1991]]). This fact that these are different can be shown by varying the problem so that these two probabilities have different numeric values. For example, assume the contestant knows that Monty does not pick the second door randomly among all legal alternatives but instead, when given an opportunity to pick between two losing doors, Monty will open the one on the right. In this situation the following two questions have different answers:▼
====Conditional probability by direct calculation====▼
# What is the probability of winning the car by ''always'' switching?▼
# What is the probability of winning the car ''given the player has picked door 1 and the host has opened door 3''?▼
The answer to the first question is 2/3, as is correctly shown by the "simple" solutions. But the answer to the second question is now different: the conditional probability the car is behind door 1 or door 2 given the host has opened door 3 (the door on the right) is 1/2. This is because Monty's preference for rightmost doors means he opens door 3 if the car is behind door 1 (which it is originally with probability 1/3) or if the car is behind door 2 (also originally with probability 1/3). For this variation, the two questions yield different answers. However as long as the initial probability the car is behind each door is 1/3, it is never to the contestant's disadvantage to switch, as the conditional probability of winning by switching is always at least 1/2. ([[#refMorganetal1991|Morgan et al. 1991]])▼
There is disagreement in the literature regarding whether vos Savant's formulation of the problem, as presented in ''Parade'' magazine, is asking the first or second question, and whether this difference is significant ([[#refRosenhouse2009|Rosenhouse 2009]]). Behrends ([[#refBehrends2008|2008]]) concludes that "One must consider the matter with care to see that both analyses are correct"; which is not to say that they are the same. One analysis for one question, another analysis for the other question. Several discussants of the paper by ([[#refMorganetal1991|Morgan et al. 1991]]), whose contributions were published alongside the original paper, strongly criticized the authors for altering vos Savant's wording and misinterpreting her intention ([[#refRosenhouse2009|Rosenhouse 2009]]). One discussant (William Bell) considered it a matter of taste whether or not one explicitly mentions that (under the standard conditions), ''which'' door is opened by the host is independent of whether or not one should want to switch.▼
Among the simple solutions, the "combined doors solution" comes closest to a conditional solution, as we saw in the discussion of approaches using the concept of odds and Bayes theorem. It is based on the deeply rooted intuition that ''revealing information that is already known does not affect probabilities''. But knowing the host can open one of the two unchosen doors to show a goat does not mean that opening a specific door would not affect the probability that the car is behind the initially chosen door. The point is, though we know in advance that the host will open a door and reveal a goat, we do not know ''which'' door he will open. If the host chooses uniformly at random between doors hiding a goat (as is the case in the standard interpretation) this probability indeed remains unchanged, but if the host can choose non-randomly between such doors then the specific door that the host opens reveals additional information. The host can always open a door revealing a goat ''and'' (in the standard interpretation of the problem) the probability that the car is behind the initially chosen door does not change, but it is ''not because'' of the former that the latter is true. Solutions based on the assertion that the host's actions cannot affect the probability that the car is behind the initially chosen appear persuasive, but the assertion is simply untrue unless each of the host's two choices are equally likely, if he has a choice ([[Monty Hall problem#refFalk1992|Falk 1992:207,213]]). The assertion therefore needs to be justified; without justification being given, the solution is at best incomplete. The answer can be correct but the reasoning used to justify it is defective.▼
Some of the confusion in the literature undoubtedly arises because the writers are using different concepts of probability, in particular, [[Bayesian probability|Bayesian]] versus [[frequentist probability]]. For the Bayesian, probability represents knowledge. For us and for the player, the car is initially equally likely to be behind each of the three doors because we know absolutely nothing about how the organizers of the show decided where to place it. For us and for the player, the host is equally likely to make either choice (when he has one) because we know absolutely nothing about how he makes his choice. These "equally likely" probability assignments are determined by symmetries in the problem. The same symmetry can be used to argue in advance that specific door numbers are irrelevant, as we saw [[Monty Hall problem#From simple to conditional by symmetry|above]].-->▼
▲The [[Monty Hall Problem#Simple Solutions|simple solutions]] above show that a player with a strategy of switching wins the car with overall probability 2/3
▲<!--====Refining the simple solution====-->
▲<!--If we assume the host opens a door at random, when given a choice, then which door the host opens gives us no information at all as to whether or not the car is behind door 1. In the simple solutions, we already observed that the probability that the car is behind door 1, the door initially chosen by the player, is initially 1/3. Moreover, the host is certainly going to open ''a'' (different) door, so opening ''a'' door (''which'' door unspecified) does not change this. 1/3 must be the average probability that the car is behind door 1 given the host picked door 2 and given the host picked door 3 because these are the only two possibilities. But these two probabilities are the same. Therefore they are both equal to 1/3 ([[#refMorganetal1991|Morgan et al. 1991]]). This shows that the chance that the car is behind door 1 given that the player initially chose this door and given that the host opened door 3 is 1/3, and it follows that the chance that the car is behind door 2 given the player initially chose door 1 and the host opened door 3 is 2/3. The analysis also shows that the overall success rate of 2/3, achieved by ''always switching'', cannot be improved, and underlines what already may well have been intuitively obvious: the choice facing the player is that between the door initially chosen, and the other door left closed by the host, the specific numbers on these doors are irrelevant.-->
[[File:Monty tree door1.svg|thumb|350px|Tree showing the probability of every possible outcome if the player initially picks Door 1]]
By definition, the [[conditional probability]] of winning by switching given the contestant initially picks door 1 and the host opens door 3 is the probability the car is behind door 2 and the host opens door 3 divided by the probability the host opens door 3. These probabilities can be determined referring to the conditional probability table below, or to an equivalent [[decision tree]] as shown to the right ([[#refChun1991|Chun 1991]]; [[#refCarlton2005|Carlton 2005]]; [[#refGrinsteadandSnell2006|Grinstead and Snell 2006:137–138]]). The conditional probability of winning by switching is (1/3)/(1/3 + 1/6), which is 2/3 {{Harv|Selvin|1975b}}.
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Many probability text books and articles in the field of probability theory derive the conditional probability solution through a formal application of [[Bayes' theorem]]; among them [[#refGill2002|Gill, 2002]] and [[#refHenze1997|Henze, 1997]]. Use of the [[odds]] form of Bayes' theorem, often called [[Bayes' rule]], makes such a derivation more transparent ([[#refRosenthal2005a|Rosenthal, 2005a]]), ([[#refRosenthal2005b|Rosenthal, 2005b]]).
Initially, the car is equally likely behind any of the three doors: the odds on door 1, door 2, and door 3 are 1:1:1. This remains the case after the player has chosen door 1, by independence. According to [[Bayes Theorem|Bayes' rule]], the posterior odds on the ___location of the car, given the host opens door 3, are equal to the prior odds multiplied by the Bayes factor or likelihood, which is by definition the probability of the new piece of information (host opens door 3) under each of the hypotheses considered (___location of the car). Now, since the player initially chose door 1, the chance the host opens door 3 is 50% if the car is behind door 1, 100% if the car is behind door 2, 0% if the car is behind door 3. Thus the Bayes factor consists of the ratios 1/2 : 1 : 0 or equivalently 1 : 2 : 0, while the prior odds were 1 : 1 : 1. Thus the posterior odds become equal to the Bayes factor 1 : 2 : 0. Given the host opened door 3, the probability the car is behind door 3 is zero, and it is twice as likely to be behind door 2
Richard Gill ([[#refGill2011a|2011]] analyzes the likelihood for the host to open door 3 as follows. Given the car is ''not'' behind door 1, it is equally likely that it is behind door 2 or 3. Therefore, the chance that the host opens door 3 is 50%. Given the car ''is'' behind door 1 the chance that the host opens door 3 is also 50%, because when the host has a choice, either choice is equally likely. Therefore, whether or not the car is behind door 1, the chance the host opens door 3 is 50%. The information "host opens door 3" contributes a Bayes factor or likelihood ratio of 1 : 1, on whether or not the car is behind door 1. Initially, the odds against door 1 hiding the car were 2 : 1. Therefore the posterior odds against door 1 hiding the car remain the same as the prior odds, 2 : 1.
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Going back to [[#refNalebuff1987|Nalebuff (1987)]], the Monty Hall problem is also much studied in the literature on [[game theory]] and [[decision theory]], and also some popular solutions correspond to this point of view. Vos Savant asks for a decision, not a chance. And the chance aspects of how the car is hidden and how an unchosen door is opened are unknown. From this point of view, one has to remember that the player has two opportunities to make choices: first of all, which door to choose initially; and secondly, whether or not to switch. Since he does not know how the car is hidden nor how the host makes choices, he may be able to make use of his first choice opportunity, as it were to neutralize the actions of the team running the quiz show, including the host.
Following [[#refGill2011|Gill, 2011]] a ''strategy'' of contestant involves two actions: the initial choice of a door and the decision to switch (or to stick) which may depend on both the door initially chosen and the door to which the host offers switching. For instance, one contestant's strategy is "choose door 1, then switch to door 2 when offered, and do not switch to door 3 when offered." Twelve such deterministic strategies of the contestant exist. Elementary comparison of contestant's strategies shows that for every strategy A there is another strategy B "pick a door then switch no matter what happens" which dominates it ([[#refGnedin2011|Gnedin, 2011]]). No matter how the car is hidden and no matter which rule the host uses when there is a choice between two goats, if A wins the car then B also does. For example, strategy A "pick door 1 then always stick with it" is dominated by the strategy B "pick door 2 then always switch after the host reveals a door": A wins when door 1 conceals the car, while B wins when one of the doors 1 and 3 conceals the car.
[[Strategic dominance]] links the Monty Hall problem to the [[game theory]]. In the [[zero-sum game]] setting of [[#refGill2011|Gill, 2011]], discarding the nonswitching strategies reduces the game to the following simple variant: the host (or the TV-team) decides on the door to hide the car, and the contestant chooses two doors (i.e., the two doors remaining after the player's first, nominal, choice). The contestant wins
▲Dominance is a strong reason to seek for a solution among always-switching strategies, under fairly general assumptions on the environment in which the contestant is making decisions. In particular, if the car is hidden by means of some randomization device – like tossing symmetric or asymmetric three-sided die – the dominance implies that a strategy maximizing the probability of winning the car will be among three always-switching strategies, namely it will be the strategy which initially picks the least likely door then switches no matter which door to switch is offered by the host.
▲[[Strategic dominance]] links the Monty Hall problem to the [[game theory]]. In the [[zero-sum game]] setting of [[#refGill2011|Gill, 2011]], discarding the nonswitching strategies reduces the game to the following simple variant: the host (or the TV-team) decides on the door to hide the car, and the contestant chooses two doors (i.e., the two doors remaining after the player's first, nominal, choice). The contestant wins (and her opponent loses) if the car is behind one of the two doors she chose.
===Solutions by Simulation===
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==History==
The earliest of several probability puzzles related to the Monty Hall problem is [[Bertrand's box paradox]], posed by [[Joseph Bertrand]] in 1889 in his ''Calcul des probabilités'' ([[#refBarbeau1993|Barbeau 1993]]). In this puzzle there are three boxes: a box containing two gold coins, a box with two silver coins, and a box with one of each. After choosing a box at random and withdrawing one coin at random that happens to be a gold coin, the question is what is the probability that the other coin is gold. As in the Monty Hall problem the intuitive answer is 1/2, but the probability is actually 2/3.
The [[Three Prisoners problem]], published in [[Martin Gardner]]'s ''Mathematical Games'' column in ''[[Scientific American]]'' in 1959 ([[#refGardner1959a|1959a]], [[#refGardner1959b|1959b]]), is equivalent to the Monty Hall problem. This problem involves three condemned prisoners, a random one of whom has been secretly chosen to be pardoned. One of the prisoners begs the warden to tell him the name of one of the others to be executed, arguing that this reveals no information about his own fate but increases his chances of being pardoned from 1/3 to 1/2. The warden obliges, (secretly) flipping a coin to decide which name to provide if the prisoner who is asking is the one being pardoned. The question is whether knowing the warden's answer changes the prisoner's chances of being pardoned. This problem is equivalent to the Monty Hall problem; the prisoner asking the question still has a 1/3 chance of being pardoned but his unnamed colleague has a 2/3 chance.
Steve Selvin posed the Monty Hall problem in a pair of letters to the ''[[The American Statistician|American Statistician]]'' in 1975 {{Harv|Selvin|1975a}}, {{Harv|Selvin|1975b}}. The first letter presented the problem in a version close to its presentation in ''Parade'' 15 years later. The second 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 offered a known lesser prize – such as $100 cash – rather than a choice to switch doors. As Monty Hall wrote to Selvin:▼
▲Steve Selvin posed the Monty Hall problem in a pair of letters to the ''[[The American Statistician|American Statistician]]'' in 1975 {{Harv|Selvin|1975a}}, {{Harv|Selvin|1975b}}. The first letter presented the problem in a version close to its presentation in ''Parade'' 15 years later. The second 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 offered a known lesser prize – such as $100 cash – rather than a choice to switch doors. As Monty Hall wrote to Selvin:{{quote|And if you ever get on my show, the rules hold fast for you – no trading boxes after the selection.|[[#refHall1975|Hall 1975]]}}
{{Quote box|width=220px|align=left|quote="You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I'll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don't need the world's highest IQ propagating more. Shame!"
- Scott Smith, Ph.D.
University of Florida}}
A restated version of Selvin's problem appeared in [[Marilyn vos Savant]]'s ''Ask Marilyn'' question-and-answer column of ''Parade'' in September 1990. {{Harv|vos Savant|1990a}} Though vos Savant gave the correct answer that switching would win two-thirds of the time, she estimates the magazine received 10,000 letters including close to 1,000 signed by PhDs, many on letterheads of mathematics and science departments, declaring that her solution was wrong. {{Harv|Tierney|1991}} Due to the overwhelming response, ''Parade'' published an unprecedented four columns on the problem. {{Harv|vos Savant|1996|p=xv}} As a result of the publicity the problem earned the alternative name Marilyn and the Goats.
In an attempt to clarify her answer she proposed a shell game {{Harv|Gardner|1982}} to illustrate: "You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you've chosen, we've learned nothing to allow us to revise the odds on the shell under your finger." She also proposed a similar simulation with three playing cards.
Despite further elaboration, many readers continued to disagree with her, but some changed their minds and agreed. Nearly 100% of those who carried out vos Savant's shell simulation changed their minds. About 56% of the general public and 71% of academics accepted the answer.
To help explain the thought process leading to the equal probability conjecture, vos Savant asked readers to consider the case where a little green woman emerges from a UFO at the point when the player has to decide whether or not to switch. The host asks the little green woman to point to one of the two unopened doors. Vos Savant noted the chance of randomly choosing the door with the prize is 1/2, because she does not know which door the player had initially chosen.
In November 1990, an equally contentious discussion of vos Savant's article took place in [[Cecil Adams]]'s column ''[[The Straight Dope]]'' ([[#refAdams1990|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's 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 2/3. 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]]'' in which Monty Hall himself was interviewed. {{Harv|Tierney|1991}} Hall appeared to understand the problem, giving the reporter a demonstration with car keys and explaining how actual game play on ''Let's Make a Deal'' differed from the rules of the puzzle.
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In an invited comment ([[#refSeymann1991|Seymann, 1991]]) and in subsequent letters to the editor, ([[#refVos_Savant1991c|vos Savant, 1991c]]; [[#RefRao1992|Rao, 1992]]; [[#refBell1992|Bell, 1992]]; [[#refHogbinandNijdam2010|Hogbin and Nijdam, 2010]]) Morgan et al. were supported by some writers, criticized by others; in each case a response by Morgan et al. is published alongside the letter or comment in ''The American Statistician''. In particular, vos Savant defended herself vigorously. Morgan et al. complained in their response to vos Savant (1991c) that vos Savant still had not actually responded to their own main point. Later in their response to Hogbin and Nijdam (2011) they did agree that it was natural to suppose that the host chooses a door to open completely at random, when he does have a choice, and hence that the conditional probability of winning by switching (i.e., conditional given the situation the player is in when he has to make his choice) has the same value, 2/3, as the unconditional probability of winning by switching (i.e., averaged over all possible situations). This equality was already emphasized by Bell (1992) who suggested that Morgan et al.'s mathematically involved solution would only appeal to statisticians, whereas the equivalence of the conditional and unconditional solutions in the case of symmetry was intuitively obvious.
When first presented with the Monty Hall problem an overwhelming majority of people assume that each door has an equal probability and conclude that switching does not matter ([[#refMueserandGranberg1999|Mueser and Granberg, 1999]]). Out of 228 subjects in one study, only 13% chose to switch ([[#refGranbergandBrown1995|Granberg and Brown, 1995:713]]). In her book ''The Power of Logical Thinking'', {{Harvtxt|vos Savant|1996|p=15}} quotes [[Cognitive psychology|cognitive psychologist]] [[Massimo Piattelli-Palmarini]] as saying "... no other statistical puzzle comes so close to fooling all the people all the time" and "that 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."
Even though most statements of the problem, notably the one in ''Parade Magazine'', do not fully specify the host's behavior or that the car's ___location is randomly selected ([[#refGranbergandBrown1995|Granberg and Brown, 1995:712]]). Krauss and Wang ([[#refKraussandWang2003|2003:10]]) conjecture that people make the standard assumptions even if they are not explicitly stated. Although
▲===Sources of confusion===
▲When first presented with the Monty Hall problem an overwhelming majority of people assume that each door has an equal probability and conclude that switching does not matter ([[#refMueserandGranberg1999|Mueser and Granberg, 1999]]). Out of 228 subjects in one study, only 13% chose to switch ([[#refGranbergandBrown1995|Granberg and Brown, 1995:713]]). In her book ''The Power of Logical Thinking'', {{Harvtxt|vos Savant|1996|p=15}} quotes [[Cognitive psychology|cognitive psychologist]] [[Massimo Piattelli-Palmarini]] as saying "... no other statistical puzzle comes so close to fooling all the people all the time" and "that 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." Pigeons repeatedly exposed to the problem show that they rapidly learn to always switch, unlike humans ([[#refHerbransonandSchroeder2010|Herbranson and Schroeder, 2010]]).
▲Although these issues are mathematically significant, even when controlling for these factors nearly all people still think each of the two unopened doors has an equal probability and conclude switching does not matter ([[#refMueserandGranberg1999|Mueser and Granberg, 1999]]). This "equal probability" assumption is a deeply rooted intuition ([[#refFalk1992|Falk 1992:202]]). People strongly tend to think probability is evenly distributed across as many unknowns as are present, whether it is or not ([[#refFoxandLevav2004|Fox and Levav, 2004:637]]). Indeed, if a player believes that sticking and switching are equally successful and therefore equally often decides to switch as to stay, they will win 50% of the time, reinforcing their original belief. Missing the unequal chances of those two doors, and in not considering that (1/3+2/3) / 2 gives a chance of 50%, similar to "the little green woman" example ([[#refSteinbach2000|Marc C. Steinbach, 2000]]).
The problem continues to attract the attention of cognitive psychologists. The typical behavior of the majority, i.e., not switching, may be explained by phenomena known in the psychological literature as: 1) the [[endowment effect]] ([[#refKahnemanetal1991|Kahneman et al., 1991]]); people tend to overvalue the winning probability of the already chosen – already "owned" – door; 2) the [[status quo bias]] ([[#refSamuelsonandZeckhauser1988|Samuelson and Zeckhauser, 1988]]); people prefer to stick with the choice of door they have already made; 3) the errors of omission vs. errors of commission effect ([[#refGilovichetal1995|Gilovich et al., 1995]]); all else considered equal, people prefer that any errors that they are responsible for to have occurred through 'omission' of taking action rather than through having taken an explicit action that later becomes known to have been erroneous. Experimental evidence confirms that these are plausible explanations which do not depend on probability intuition ([[#refKaivantoetal2014|Kaivanto et al., 2014]]; [[#refMoroneandFiore2007|Morone and Fiore, 2007]]).
▲===Criticism of the simple solutions===
▲As already remarked, most sources in the field of [[probability]], including many introductory probability textbooks, solve the problem by showing the [[conditional probabilities]] the car is behind door 1 and door 2 are 1/3 and 2/3 (not 1/2 and 1/2) given the contestant initially picks door 1 and the host opens door 3; various ways to derive and understand this result were given in the previous subsections.
▲Among these sources are several that explicitly criticize the popularly presented "simple" solutions, saying these solutions are "correct but ... shaky" ([[#refRosenthal2005a|Rosenthal 2005a]]), or do not "address the problem posed" ([[#refGillman1992|Gillman 1992]]), or are "incomplete" ([[#refLucasetal2009|Lucas et al. 2009]]), or are "unconvincing and misleading" ([[#refEisenhauer2001|Eisenhauer 2001]]) or are (most bluntly) "false" ([[#refMorganetal1991|Morgan et al. 1991]]).
▲Some say that these solutions answer a slightly different question – one phrasing is "you have to announce ''before a door has been opened'' whether you plan to switch" ([[#refGillman1992|Gillman 1992]], emphasis in the original).
▲The simple solutions show in various ways that a contestant who is determined to switch will win the car with probability 2/3, and hence that switching is the winning strategy, if the player has to choose in advance between "always switching", and "always staying". However, the probability of winning by ''always'' switching is a logically distinct concept from the probability of winning by switching ''given the player has picked door 1 and the host has opened door 3''. As one source says, "the distinction between [these questions] seems to confound many" ([[#refMorganetal1991|Morgan et al. 1991]]). This fact that these are different can be shown by varying the problem so that these two probabilities have different numeric values. For example, assume the contestant knows that Monty does not pick the second door randomly among all legal alternatives but instead, when given an opportunity to pick between two losing doors, Monty will open the one on the right. In this situation the following two questions have different answers:
▲# What is the probability of winning the car by ''always'' switching?
▲# What is the probability of winning the car ''given the player has picked door 1 and the host has opened door 3''?
▲The answer to the first question is 2/3, as is correctly shown by the "simple" solutions. But the answer to the second question is now different: the conditional probability the car is behind door 1 or door 2 given the host has opened door 3 (the door on the right) is 1/2. This is because Monty's preference for rightmost doors means he opens door 3 if the car is behind door 1 (which it is originally with probability 1/3) or if the car is behind door 2 (also originally with probability 1/3). For this variation, the two questions yield different answers. However as long as the initial probability the car is behind each door is 1/3, it is never to the contestant's disadvantage to switch, as the conditional probability of winning by switching is always at least 1/2. ([[#refMorganetal1991|Morgan et al. 1991]])
▲There is disagreement in the literature regarding whether vos Savant's formulation of the problem, as presented in ''Parade'' magazine, is asking the first or second question, and whether this difference is significant ([[#refRosenhouse2009|Rosenhouse 2009]]). Behrends ([[#refBehrends2008|2008]]) concludes that "One must consider the matter with care to see that both analyses are correct"; which is not to say that they are the same. One analysis for one question, another analysis for the other question. Several discussants of the paper by ([[#refMorganetal1991|Morgan et al. 1991]]), whose contributions were published alongside the original paper, strongly criticized the authors for altering vos Savant's wording and misinterpreting her intention ([[#refRosenhouse2009|Rosenhouse 2009]]). One discussant (William Bell) considered it a matter of taste whether or not one explicitly mentions that (under the standard conditions), ''which'' door is opened by the host is independent of whether or not one should want to switch.
▲Among the simple solutions, the "combined doors solution" comes closest to a conditional solution, as we saw in the discussion of approaches using the concept of odds and Bayes theorem. It is based on the deeply rooted intuition that ''revealing information that is already known does not affect probabilities''. But knowing the host can open one of the two unchosen doors to show a goat does not mean that opening a specific door would not affect the probability that the car is behind the initially chosen door. The point is, though we know in advance that the host will open a door and reveal a goat, we do not know ''which'' door he will open. If the host chooses uniformly at random between doors hiding a goat (as is the case in the standard interpretation) this probability indeed remains unchanged, but if the host can choose non-randomly between such doors then the specific door that the host opens reveals additional information. The host can always open a door revealing a goat ''and'' (in the standard interpretation of the problem) the probability that the car is behind the initially chosen door does not change, but it is ''not because'' of the former that the latter is true. Solutions based on the assertion that the host's actions cannot affect the probability that the car is behind the initially chosen appear persuasive, but the assertion is simply untrue unless each of the host's two choices are equally likely, if he has a choice ([[Monty Hall problem#refFalk1992|Falk 1992:207,213]]). The assertion therefore needs to be justified; without justification being given, the solution is at best incomplete. The answer can be correct but the reasoning used to justify it is defective.
▲Some of the confusion in the literature undoubtedly arises because the writers are using different concepts of probability, in particular, [[Bayesian probability|Bayesian]] versus [[frequentist probability]]. For the Bayesian, probability represents knowledge. For us and for the player, the car is initially equally likely to be behind each of the three doors because we know absolutely nothing about how the organizers of the show decided where to place it. For us and for the player, the host is equally likely to make either choice (when he has one) because we know absolutely nothing about how he makes his choice. These "equally likely" probability assignments are determined by symmetries in the problem. The same symmetry can be used to argue in advance that specific door numbers are irrelevant, as we saw [[Monty Hall problem#From simple to conditional by symmetry|above]].
==Variants==
The table below shows a variety of possible host behaviors, other than those specified in the standard assumptions, and the resulting probability of success by switching.
▲===Other host behaviors===
Morgan et al. ([[#refMorganetal1991|1991]]) and Gillman ([[#refGillman1992|1992]]) both show a more general solution where the car is (uniformly) randomly placed but the host is not constrained to pick uniformly randomly if the player has initially selected the car, which is how they both interpret the statement of the problem in ''Parade'' despite the author's disclaimers. Both changed the wording of the ''Parade'' version to emphasize that point when they restated the problem. They consider a scenario where the host chooses between revealing two goats with a preference expressed as a probability ''q'', having a value between 0 and 1. If the host picks randomly ''q'' would be 1/2 and switching wins with probability 2/3 regardless of which door the host opens. If the player picks door 1 and the host's preference for door 3 is ''q'', then the probability the host opens door 3 and the car is behind door 2 is 1/3 while the probability the host opens door 3 and the car is behind door 1 is (1/3)''q''. These are the only cases where the host opens door 3, so the conditional probability of winning by switching ''given the host opens door 3'' is (1/3)/(1/3 + (1/3)''q'') which simplifies to 1/(1+''q''). Since ''q'' can vary between 0 and 1 this conditional probability can vary between 1/2 and 1. This means even without constraining the host to pick randomly if the player initially selects the car, the player is never worse off switching. However neither source suggests the player knows what the value of ''q'' is so the player cannot attribute a probability other than the 2/3 that vos Savant assumed was implicit.▼
<div style="text-align:center;">
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▲Morgan et al. ([[#refMorganetal1991|1991]]) and Gillman ([[#refGillman1992|1992]]) both show a
===''N''-doors===
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* In 2009 a book-length discussion of the problem, its history, methods of solution, and variations, was published by Oxford University Press ([[#refRosenhouse2009|Rosenhouse 2009]]). <!-- not outside of academia --><!--So what-->
* The problem is presented, discussed, and tested in the television show ''[[MythBusters]]'' on 23 November 2011. This paradox was not only tested to see if there was an advantage to switching vs. sticking (which, in a repeated sample of 49 "tests", showed a significant advantage to switching), but they also tested the behavior of "contestants" presented with the same situation. All 20 of the common "contestants" tested chose to stay with their original choice.
* The problem was also discussed and tested on the television show ''[[James May's Man Lab]]'' on 11 April 2013. In this presentation, each test was done by presenting three identical beer cans, two of which had been shaken (with the result that opening it would douse the person in beer foam). James May performed this test 100 times, each time switching his choice from his original choice after one of the shaken cans was removed. In the end, he was doused 40 times, while his colleague Sim, who had to pick the remaining beer can, was doused 60 times. The resulting percentage was roughly what they expected.<ref>
* Craig Whitaker's actual letter to vos Savant has been found, and his original question reported in Morgan et al. Response to Hogbin and Nijdam (2011): "I've worked out two different situations (based on Monty's prior behavior i.e. weather [sic.] or not he knows what's behind the doors) in one situation it is to your advantage to switch, in the other there is no advantage to switch." "What do you think?"
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==References==
{{refbegin |colwidth=30em}}
* {{cite journal |authorlink= Cecil Adams |last= Adams |first= Cecil |ref= refAdams1990 |url= http://www.straightdope.com/classics/a3_189.html |title= 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? |journal= The Straight Dope |date= 2 November 1990 |accessdate= 25 July 2005 }}
* {{cite journal |last= Barbeau |first= Edward |ref= refBarbeau1993 |year= 1993 |title= Fallacies, Flaws, and Flimflam: The Problem of the Car and Goats |journal= The College Mathematics Journal |volume= 24 |issue= 2 |pages= 149–154|doi= 10.1080/07468342.1993.11973519 }}
* {{cite book |last= Barbeau |first= Edward |ref= refBarbeau2000 |year= 2000 |title= Mathematical Fallacies, Flaws and Flimflam |publisher= The Mathematical Association of America |isbn= 0-88385-529-1}}
* {{cite book |ref= refBehrends2008 |title= Five-Minute Mathematics |last= Behrends |first= Ehrhard |publisher= AMS Bookstore |year= 2008 |isbn= 978-0-8218-4348-2 |page= 57 |url=
* {{cite journal |last= Bell |first= William |ref= refBell1992 |title= Comment on 'Let's make a deal' by Morgan et al. |journal= American Statistician |volume= 46 |issue= 3 |page= 241 |date= August 1992}}
* {{cite web |ref=refBloch2008 |url=http://www.andybloch.com/gl/pub/article.php?story=2008031308241327 |title=21: The Movie (my review) |first=Andy |last=Bloch |authorlink=Andy Bloch |year=2008 |accessdate=2008-05-05 }}
* {{cite journal |ref=refCarlton2005 |url=http://www.amstat.org/publications/JSE/v13n2/carlton.html |title=Pedigrees, Prizes, and Prisoners: The Misuse of Conditional Probability |first=Matthew |last=Carlton |year=2005 |journal=Journal of Statistics Education [online] |volume=13 |issue=2 |accessdate=2010-05-29 }}
* {{cite journal |last= Chun |first= Young H. |ref= refChun1991 |year= 1991 |title= Game Show Problem |journal= OR/MS Today |volume= 18 |issue= 3 |page= 9}}
* {{cite web |last1= D'Ariano |first1= G.M. |last2= Gill |first2= R.D. |last3= Keyl |first3= M. |last4= Werner |first4= R.F. |last5= Kümmerer |first5= K. |last6= Maasen |first6= H. |
* {{cite web |ref=refDevlin2003 |url=http://www.maa.org/external_archive/devlin/devlin_07_03.html |title=Devlin's Angle: Monty Hall |publisher=The Mathematical Association of America |first=Keith |last=Devlin |authorlink=Keith Devlin |date=July–August 2003 |accessdate=23 June 2014 }}
* {{cite web |ref=refDevlin2005 |url=http://www.maa.org/external_archive/devlin/devlin_12_05.html |title=Devlin's Angle: Monty Hall revisited |publisher=The Mathematical Association of America |first=Keith |last=Devlin |authorlink=Keith Devlin |date=December 2005 |accessdate=23 June 2014 }}
* {{cite news |ref= refEconomist1999 |work= The Economist |title= Getting the goat: When it comes to weighing risks and probabilities, keep in mind this golden rule: never trust your guesses |volume= 350 |page= 110 |url= http://www.economist.com/node/187166 |date= 18 February 1999 }}
* {{cite book |last= Diaconis |first= Persi |ref= refDiaconis1988 |year= 1988 |title= Group representations in probability and statistics |series= IMS Lecture Notes |publisher= Institute of Mathematical Statistics |isbn= 0-940600-14-5}}
* {{cite journal |last= Eisenhauer |first= Joseph G. |ref= refEisenhauer2001 |year= 2001 |title= The Monty Hall Matrix |journal= Teaching Statistics |volume= 22 |issue= 1 |pages= 17–20 |url= http://isds.bus.lsu.edu/chun/teach/reading-a/matrix.pdf
* {{cite journal |last= Falk |first= Ruma |ref= refFalk1992 |year= 1992 |title= A closer look at the probabilities of the notorious three prisoners |journal= Cognition |volume= 43 |issue= 3 |pages= 197–223 |doi=10.1016/0010-0277(92)90012-7|pmid= 1643813 }}
* {{cite journal |last1= Flitney |first1= Adrian P. |authorlink2= Derek Abbott |last2= Abbott |first2= Derek |lastauthoramp= yes |ref= refFlitney2002 |year=2002 |title= Quantum version of the Monty Hall problem |journal= Physical Review A |volume= 65 |issue= 6 |page= 062318 |doi= 10.1103/PhysRevA.65.062318 |id= Art. No. 062318, 2002 |arxiv= quant-ph/0109035}}
* {{cite journal |last1= Fox |first1= Craig R. |last2= Levav |first2= Jonathan |lastauthoramp= yes |ref= refFoxandLevav2004 |year= 2004 |title= Partition-Edit-Count: Naive Extensional Reasoning in Judgment of Conditional Probability |journal= Journal of Experimental Psychology: General |volume= 133 |issue= 4 |pages= 626–642|doi= 10.1037/0096-3445.133.4.626 |pmid= 15584810 }}
* {{cite web |date=18 September 2012 |first=Caroline |last=Frost |title=TV Review: Derren Brown: Svengali – How Does He Do It? No, Really – I'm Asking... |url=http://www.huffingtonpost.co.uk/2012/09/18/tv-review-derren-brown-svengali_n_1894718.html
* {{cite journal |authorlink= Martin Gardner |last= Gardner |first= Martin |ref= refGardner1959a |title= Mathematical Games |journal= Scientific American |date= October 1959a |pages= 180–182 |postscript=. Reprinted in ''The Second Scientific American Book of Mathematical Puzzles and Diversions''}}
* {{cite journal |last= Gardner |first= Martin |ref= refGardner1959b |title= Mathematical Games |journal= Scientific American |date= November 1959b |page= 188}}
* {{cite book |last=Gardner |first=Martin |year=1982 |title=Aha! Gotcha: Paradoxes to Puzzle and Delight |publisher=W. H. Freeman |isbn=978-0716713616
* {{cite book |authorlink= Jeff Gill |last= Gill |first= Jeff |year= 2002 |title= Bayesian Methods |pages= 8–10 |publisher= CRC Press |isbn= 1-58488-288-3 |postscript=. ({{Google books|IJ3XTLQViM4C|restricted online copy|page=8}})}}
* {{cite book |authorlink= Richard D. Gill |last= Gill |first= Richard |ref= refGill2010 |chapter= Monty Hall problem |pages=858–863 |title= International Encyclopaedia of Statistical Science |publisher= Springer |year= 2010 |arxiv= 1002.3878v2 }}
* {{cite journal |last= Gill |first= Richard |ref= refGill2011 |title= The Monty Hall Problem is not a probability puzzle (it's a challenge in mathematical modelling) |journal= Statistica Neerlandica |volume= 65 |issue= 1 |pages= 58–71 |date= February 2011 |arxiv= 1002.0651v3 |doi=10.1111/j.1467-9574.2010.00474.x}}
* {{cite web |ref= refGill2011a |last= Gill |first= Richard |year= 2011 |title= The Monty Hall Problem |publisher= Mathematical Institute, University of Leiden, Netherlands |pages= 10–13 |date= 17 March 2011a |url= http://www.math.leidenuniv.nl/~gill/mhp-statprob.pdf
* {{cite
* {{cite journal |ref= refGillman1992 |authorlink= Leonard Gillman |last= Gillman |first= Leonard |year= 1992 |jstor= 2324540 |title= The Car and the Goats |journal= American Mathematical Monthly |volume= 99 |issue= 1 |pages= 3–7|doi= 10.1080/00029890.1992.11995797 }}
* {{cite journal|ref=refGilovichetal1995|last1=Gilovich|first1=T.|last2=Medvec|first2=V.H.|last3=Chen|first3=S.|lastauthoramp=yes|title=Commission, Omission, and Dissonance Reduction: Coping with Regret in the "Monty Hall" Problem|journal=Personality and Social Psychology Journal|year=1995|volume=21|issue=2|pages=182–190|doi=10.1177/0146167295212008|url=http://psp.sagepub.com/content/21/2/182.abstract}}
* {{cite web |ref= refGnedin2011 |authorlink= Sasha Gnedin |last= Gnedin |first= Sasha |year= 2011 |title= The Mondee Gills Game |work= The Mathematical Intelligencer |url= http://www.springerlink.com/content/8402812734520774/fulltext.pdf }}{{Dead link|date=July 2021 |bot=InternetArchiveBot |
* {{cite book |ref= refGranberg1996 |last= Granberg |first= Donald |year= 1996 |chapter= To Switch or Not to Switch |editor-last= vos Savant |editor-first= Marilyn |title= The Power of Logical Thinking |publisher= St. Martin's Press |isbn= 0-312-30463-3 |postscript=. ({{Google books|jNndlc2W4pAC|restricted online copy |page=169}})}}
* {{cite journal |last1= Granberg |first1= Donald |last2= Brown |first2= Thad A. |lastauthoramp= yes |ref= refGranbergandBrown1995 |year= 1995 |title= The Monty Hall Dilemma |journal= Personality and Social Psychology Bulletin |volume= 21 |issue= 7 |pages= 711–729|doi= 10.1177/0146167295217006 }}
* {{cite book |ref= refGrinsteadandSnell2006 |last1= Grinstead |first1= Charles M. |last2= Snell |first2= J. Laurie |lastauthoramp= yes |title= Grinstead and Snell's Introduction to Probability |url= http://www.math.dartmouth.edu/~prob/prob/prob.pdf |accessdate= 2 April 2008 |date= 4 July 2006
* {{cite book |ref= refGruber2010 |title= The World's 200 Hardest Brain Teasers |last= Gruber |first= Gary |publisher= Sourcebooks
* {{cite book |ref=Hayden2009 |title=This Book Does Not Exist: Adventures in the Paradoxical |year=2009 |publisher= Fall River Press |___location=New York |isbn=978-1-4351-1071-7 |pages=90–93 |first1= Gary |last1= Hayden |first2= Michael |last2= Picard |lastauthoramp= yes |edition=2009 |chapter=5: Probability Paradoxes |quote=In September of 1990, in an issue of Parade magazine, High-IQ columnist Marilyn vos Savant introduced this puzzle in the weekly "Ask Marilyn" section. The puzzle was based on the TV show, Let's Make a Deal, in which host Monty Hall presented contestants with essentially the same choices in order to win either a high-value prize or one of two undesirable alternatives...Marilyn's analysis revealed, quite correctly, that switching doors doubles the chances of driving away [in] a shiny new car. But readers wrote in their thousands to disagree. Many of her severest critics were people with math and science Ph.D.s, who hauled her over the coals for what they considered a monumental gaffe. The debate raged for almost a year, culminating in a front-page article in the New York Times on Sunday, July 21, 1991, in which Marilyn's analysis was vindicated.}}
* {{cite web |ref= refHall1975 |authorlink= Monty Hall |last= Hall |first= Monty |year= 1975 |url= http://www.letsmakeadeal.com/problem.htm |title= The Monty Hall Problem |work= LetsMakeADeal.com |accessdate= 15 January 2007 |postscript= . Includes 12 May 1975 letter to Steve Selvin }}
* {{cite book |ref= refHenze1997 |last= Henze |first= Norbert |origyear= 1997 |title= Stochastik für Einsteiger: Eine Einführung in die faszinierende Welt des Zufalls |edition= 9th |year= 2011 |pages= 50–51, 105–107 |publisher= Springer |isbn= 9783834818454 |postscript=. ({{Google books|esknPQ4sUs4C|restricted online copy|page=105}})}}
* {{cite journal |ref= refHerbransonandSchroeder2010 |last1= Herbranson |first1= W. T. |last2= Schroeder |first2= J. |lastauthoramp= yes |year= 2010 |title= Are birds smarter than mathematicians? Pigeons (''Columba livia'') perform optimally on a version of the Monty Hall Dilemma |journal= Journal of Comparative Psychology |volume= 124 |issue= 1 |pages= 1–13 |
*{{cite journal |ref= refHogbinandNijdam2010 |
* {{cite journal |ref= refKahnemanetal1991 |authorlink= Daniel Kahneman |
* {{cite journal|ref=refKaivantoetal2014|last1=Kaivanto|first1=K.|last2=Kroll|first2=E.B.|last3=Zabinski|first3=M.|lastauthoramp=yes|title=Bias Trigger Manipulation and Task-Form Understanding in Monty Hall|journal=Economics Bulletin|year=2014|volume=34|issue=1|pages=89–98|url=http://www.accessecon.com/Pubs/EB/2014/Volume34/EB-14-V34-I1-P10.pdf}}
* {{cite journal |ref= refKraussandWang2003 |last1= Krauss |first1= Stefan |last2= Wang |first2= X. T. |lastauthoramp= yes |year= 2003 |title= The Psychology of the Monty Hall Problem: Discovering Psychological Mechanisms for Solving a Tenacious Brain Teaser |journal= Journal of Experimental Psychology: General |volume= 132 |issue= 1 |pages= 3–22 |doi= 10.1037/0096-3445.132.1.3 |pmid= 12656295 |url= http://www.diss.fu-berlin.de/diss/servlets/MCRFileNodeServlet/FUDISS_derivate_000000001141/04_chap2.pdf
* {{cite journal |ref= refLucasetal2009 |last1= Lucas |first1= Stephen |first2= Jason |last2= Rosenhouse |first3= Andrew |last3= Schepler |lastauthoramp= yes |year= 2009 |title= The Monty Hall Problem, Reconsidered |journal= Mathematics Magazine |volume= 82 |issue= 5 |pages= 332–342 |doi= 10.4169/002557009X478355 |url= http://educ.jmu.edu/~lucassk/Papers/MHOverview2.pdf
* {{cite book |ref= refMack1992 |title= The Unofficial IEEE Brainbuster Gamebook |last= Mack |first= Donald R. |publisher= Wiley-IEEE |year= 1992 |isbn= 978-0-7803-0423-9 |page= 76 |url=
* {{cite book |ref=refMagliozziandMagliozzi1998 |last1= Magliozzi |first1= Tom |last2= Magliozzi |first2= Ray |lastauthoramp= yes |authorlink=Tom Magliozzi |title= Haircut in Horse Town: & Other Great Car Talk Puzzlers |publisher= Diane Pub Co. |year=1998 |isbn=0-7567-6423-8}}
* {{cite book |ref= refMartin1989 |last= Martin |first= Phillip |url= http://sites.google.com/site/psmartinsite/Home/bridge-articles/the-monty-hall-trap |chapter= The Monty Hall Trap |origyear= 1989 |editor-last1= Granovetter |editor1-first= Pamela |editor2-last= Granovetter |editor2-first= Matthew |year= 1993 |
* {{cite book |ref= refMartin2002 |title= There are two errors in the {{not a typo |the}} title of this book |last= Martin |first= Robert M. |edition= 2nd |publisher= Broadview Press |year= 2002 |isbn= 978-1-55111-493-4 |pages= 57–59 |url=
* {{cite book |ref= CITEREFMlodinow2008 |last= Mlodinow |first= Leonard |title= The Drunkard's Walk: How Randomness Rules Our Lives |year= 2008 |pages= 53–56 |postscript=. {{cite AV media |url= http://www.youtube.com/watch?feature=player_detailpage&v=zAt27VwavJ8&t=1500 |title= The Monty Hall Problem |via= YouTube |time= 25:00–28:00}} }}
* {{cite journal |ref= refMorganetal1991 |last1= Morgan |first1= J. P. |last2= Chaganty |first2= N. R. |last3= Dahiya |first3= R. C. |last4= Doviak |first4= M. J. |lastauthoramp= yes |year= 1991 |jstor= 2684453 |title= Let's make a deal: The player's dilemma |journal= American Statistician |volume= 45 |issue= 4 |pages= 284–287 |doi=10.1080/00031305.1991.10475821}}
* {{cite web |ref= refMoroneandFiore2007 |last1= Morone |first1= A. |first2= A. |last2= Fiore |lastauthoramp= yes |year= 2007 |title= Monty Hall's Three Doors for Dummies |publisher= Dipartimento di Scienze Economiche e Metodi Matematici – Università di Bari, Southern Europe Research in Economic Studies – S.E.R.I.E.S. |id= Working Paper no. 0012 |url= http://ideas.repec.org/p/bai/series/wp0012.html }}
* {{cite web |ref= refMueserandGranberg1999 |last1= Mueser |first1= Peter R. |last2= Granberg |first2= Donald |lastauthoramp= yes |date= May 1999 |url= http://econpapers.repec.org/paper/wpawuwpex/9906001.htm |title= The Monty Hall Dilemma Revisited: Understanding the Interaction of Problem Definition and Decision Making |publisher= University of Missouri |id= Working Paper 99-06 |accessdate= 10 June 2010 }}
* {{cite journal |ref= refNalebuff1987 |authorlink= Barry Nalebuff |last= Nalebuff |first= Barry |title= Puzzles: Choose a Curtain, Duel-ity, Two Point Conversions, and More |journal= Journal of Economic Perspectives |volume= 1 |issue= 2 |pages= 157–163 |date= Autumn 1987 |doi=10.1257/jep.1.2.157}}
* {{cite journal |ref= CITEREFRao1992 |last= Rao |first= M. Bhaskara |title= Comment on ''Let's make a deal'' by Morgan et al. |journal= American Statistician |volume= 46 |issue= 3 |pages= 241–242 |date= August 1992}}
* {{cite book |ref= refRosenhouse2009 |last= Rosenhouse |first= Jason |title= The Monty Hall Problem |publisher= Oxford University Press |year= 2009 |isbn= 978-0-19-536789-8}}
* {{cite journal |ref= refRosenthal2005a |title= Monty Hall, Monty Fall, Monty Crawl |first= Jeffrey S. |last= Rosenthal |date= September 2005a |journal= Math Horizons |pages= 5–7 |url= http://probability.ca/jeff/writing/montyfall.pdf
* {{cite book |ref= refRosenthal2005b |last= Rosenthal |first= Jeffrey S. |year= 2005b |title= Struck by Lightning: the Curious World of Probabilities |publisher= Harper Collins |isbn= 978-0-00-200791-7}}
* {{cite journal |ref= refSamuelsonandZeckhauser1988 |last1= Samuelson |first1= W |first2= R. |last2= Zeckhauser |lastauthoramp=yes |year= 1988 |title= Status quo bias in decision making |journal= Journal of Risk and Uncertainty |volume= 1 |pages= 7–59 |doi=10.1007/bf00055564}}
* {{cite book |ref= refSchwager1994 |title= The New Market Wizards |last= Schwager |first= Jack D. |publisher= Harper Collins |year= 1994 |isbn= 978-0-88730-667-9 |page= 397 |url=
* {{cite journal |ref= CITEREFSelvin1975a |last=Selvin |first=Steve |year=1975a |title=A problem in probability (letter to the editor) |journal=American Statistician |volume=29 |issue=1 |page= 67 |date= February 1975 |issn= |jstor= 2683689 }}
* {{cite journal |ref= CITEREFSelvin1975b |last=Selvin |first=Steve |year=1975b |title=On the Monty Hall problem (letter to the editor) |journal=American Statistician |volume=29 |issue=3 |page=134 |date= August 1975 |issn= |jstor= 2683443 }}
* {{cite journal |ref= CITEREFSeymann1991 |last= Seymann |first= R. G. |year= 1991 |jstor= 2684454 |title= Comment on Let's make a deal: The player's dilemma |journal= American Statistician |volume= 45 |pages= 287–288}}
* {{cite web |ref= refSteinbach2000 |last= Steinbach |first= Marc C. |year= 2000 |url= http://www.zib.de/Publications/Reports/ZR-00-40.pdf
* {{cite journal |ref= refStibeletal2008 |authorlink= Jeff Stibel |last1= Stibel |first1= Jeffrey |last2= Dror |first2= Itiel |last3= Ben-Zeev |first3= Talia |year= 2008 |title= The Collapsing Choice Theory: Dissociating Choice and Judgment in Decision Making |journal= Theory and Decision |url= http://users.ecs.soton.ac.uk/id/TD%20choice%20and%20judgment.pdf
* {{cite news |ref= CITEREFTierney1991 |last= Tierney |first= John |authorlink= John Tierney (journalist) |url= http://query.nytimes.com/gst/fullpage.html?res=9D0CEFDD1E3FF932A15754C0A967958260 |title= Behind Monty Hall's Doors: Puzzle, Debate and Answer? |newspaper= The New York Times |date= 21 July 1991 |accessdate= 18 January 2008 }}
* {{cite news |ref= CITEREFTierney2008 |last= Tierney |first= John |url= http://www.nytimes.com/2008/04/08/science/08tier.html |title= And Behind Door No. 1, a Fatal Flaw |journal= The New York Times |date= 8 April 2008 |accessdate= 8 April 2008 }}
* {{cite journal |ref= refVazsonyi1999 |last= Vazsonyi |first= Andrew |title= Which Door Has the Cadillac? |journal= Decision Line |date= December 1998 – January 1999 |pages= 17–19 |url= http://www.decisionsciences.org/DecisionLine/Vol30/30_1/vazs30_1.pdf
* {{cite journal |ref= refvosSavantGSP |author1-link= Marilyn vos Savant |last= vos Savant |first= Marilyn |date= 1990–91 |title= Game Show Problem |url= http://marilynvossavant.com/game-show-problem/ |accessdate= 16 December 2012 |journal= |archive-date= 29 April 2012 |archive-url= https://web.archive.org/web/20120429013941/http://marilynvossavant.com/game-show-problem/ |url-status= dead }}
* {{cite journal |last=vos Savant |first=Marilyn |ref=CITEREFvos_Savant1990a |url=http://marilynvossavant.com/game-show-problem/ |title=Ask Marilyn |journal=Parade Magazine |page=16 |date=9 September 1990a |access-date=8 July 2014 |archive-date=29 April 2012 |archive-url=https://web.archive.org/web/20120429013941/http://marilynvossavant.com/game-show-problem/ |url-status=dead }}
* {{cite journal |last= vos Savant |first= Marilyn |ref= CITEREFvos_Savant1990b |url= http://marilynvossavant.com/game-show-problem/ |title= Ask Marilyn |journal= Parade Magazine |page= 25 |date= 2 December 1990b |access-date= 8 July 2014 |archive-date= 29 April 2012 |archive-url= https://web.archive.org/web/20120429013941/http://marilynvossavant.com/game-show-problem/ |url-status= dead }}
* {{cite journal |last= vos Savant |first= Marilyn |ref= CITEREFvos_Savant1991a |url= http://marilynvossavant.com/game-show-problem/ |title= Ask Marilyn |journal= Parade Magazine |page= 12 |date= 17 February 1991a |access-date= 8 July 2014 |archive-date= 29 April 2012 |archive-url= https://web.archive.org/web/20120429013941/http://marilynvossavant.com/game-show-problem/ |url-status= dead }}
* {{cite journal |last= vos Savant |first= Marilyn |ref= CITEREFvos_Savant1991b |url= http://marilynvossavant.com/game-show-problem/ |title= Ask Marilyn |journal= Parade Magazine |page= 26 |date= 7 July 1991b |access-date= 8 July 2014 |archive-date= 29 April 2012 |archive-url= https://web.archive.org/web/20120429013941/http://marilynvossavant.com/game-show-problem/ |url-status= dead }}
* {{cite journal |last= vos Savant |first= Marilyn |ref= CITEREFvos_Savant1991c |title= Marilyn vos Savant's reply |department= Letters to the editor |journal= American Statistician |volume= 45 |issue= 4 |page= 347 |date= November 1991c}}
* {{cite book
* {{cite journal |last= vos Savant |first= Marilyn |ref= CITEREFvos_Savant2006 |title= Ask Marilyn |journal= Parade Magazine |page= 6 |date= 26 November 2006}}
* {{cite web |ref=refWilliams2004 |url=http://www.nd.edu/~rwilliam/stats1/appendices/xappxd.pdf |title=Appendix D: The Monty Hall Controversy |first=Richard |last=Williams |year=2004
* {{cite book |ref= refWheeler1991 |title= Phylogenetic analysis of DNA sequences |chapter= Congruence Among Data Sets: A Bayesian Approach |last= Wheeler |first= Ward C. |editor-first1= Michael M. |editor-last1= Miyamoto |editor-first2= Joel |editor-last2= Cracraft |lastauthoramp= yes |publisher= Oxford University Press US |year= 1991 |isbn= 978-0-19-506698-2 |page= 335 |chapter-url=
* {{cite journal |last= Whitaker |first= Craig F. |ref= refWhitaker1990 |title= [Formulation by Marilyn vos Savant of question posed in a letter from Craig Whitaker]. Ask Marilyn |journal= Parade Magazine |page= 16 |date= 9 September 1990}}
<!-- {{cite journal |first= Marilyn |last= vos Savant |date= November 26 – December 2, 2006 |title= Ask Marilyn |journal= Parade Classroom Teacher's Guide |pages= 3 |url= http://www.paradeclassroom.com/tg_folders/2006/1126/TG_11262006.pdf |format= [[PDF]] |accessdate= 27 November 2006 |isbn= 0-312-08136-7 }} -->
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* [http://www.marilynvossavant.com/articles/gameshow.html The Game Show Problem]–the original question and responses on Marilyn vos Savant's web site
* [http://math.ucsd.edu/~crypto/Monty/Montytitle.html University of California San Diego, Monty Knows Version and Monty Does Not Know Version, An Explanation of the Game]
* "[http://demonstrations.wolfram.com/MontyHallParadox/ Monty Hall Paradox]" by Matthew R. McDougal, [[The Wolfram Demonstrations Project]] (simulation)
* [http://www.khanacademy.org/math/probability/v/monty-hall-problem
* [http://www.nytimes.com/2008/04/08/science/08monty.html The Monty Hall Problem] at The New York Times (simulation)
* [http://www.matifutbol.com/en/reserve.html The reserve player's chance] A practical example on the Monty Hall paradox.
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{{DEFAULTSORT:Monty Hall Problem}}
[[:Category:Decision
[[:Category:Game theory]]
[[:Category:Let's Make a Deal]]
[[:Category:Mathematical problems]]
[[:Category:Microeconomics]]
[[:Category:Probability theory paradoxes]]
[[:Category:Named probability problems]]
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