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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.]]
==Untitled==
The '''Monty Hall problem''' is a [[probability]] puzzle loosely based on the American television game show ''[[Let's Make a Deal]]'' and named after the show's original host, [[Monty Hall]]. The problem, also called the '''Monty Hall paradox''', is a [[paradox#Logical paradox|''veridical paradox'']] because the result appears odd but is demonstrably true. The Monty Hall problem, in its usual interpretation, is mathematically equivalent to the earlier [[Three Prisoners problem]], and both bear some similarity to the much older [[Bertrand's box paradox]].
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|width2=197
|caption2=Player's pick remains a 1/3 chance, while the other two doors a combined 2/3 chance, 2/3 for the still unopened one and 0 for the one the host opened.}}
The solution presented by vos Savant in ''Parade'' ([[#refvosSavant1990b|vos Savant 1990b]]) shows the three possible arrangements of one car and two goats behind three doors and the result of switching or staying after initially picking Door 1 in each case:
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[[File:Monty tree door1.svg|thumb|350px|Tree showing the probability of every possible outcome if the player initially picks Door 1]]
==Aids to understanding==▼
===Other solutions===▼
An even simpler solution is to reason that switching loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3. ([[#refCarlton2005|Carlton 2005]]).▼
Another way to understand the solution is to consider the two original unchosen doors together. 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 choose the opened door ([[#refAdams1990|Adams 1990]]; [[#refDevlin2003|Devlin 2003]]; [[#refWilliams2004|Williams 2004]]; [[#refStibeletal2008|Stibel et al., 2008]]).▼
As [[Cecil Adams]] puts it ([[#refAdams1990|Adams 1990]]), "Monty is saying in effect: you can keep your one door or you can have the other two doors." The player therefore has the choice of either sticking with the original choice of door, or choosing the sum of the contents of the two other doors, as the 2/3 chance of hiding the car has not been changed by the opening of one of these doors.▼
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.{{' "}}▼
▲==Aids to understanding==
▲===Simulation===
[[File:Monty problem monte carlo.svg|thumb|Simulation of 30 outcomes of the Monty Hall problem]]
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Another simulation, suggested by vos Savant, employs the "host" hiding a penny, representing the car, under one of three cups, representing the doors; or hiding a pea under one of three shells.
That switching has a probability of 2/3 of winning the car runs counter to many people's intuition. If there are two doors left, then why isn't each door 1/2? The intuition may be aided by generalizing the problem to have a large number of doors so that the player's initial choice has a small chance of winning.
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In addition to the "equal probability" intuition, a competing and deeply rooted intuition is that ''revealing information that is already known does not affect probabilities''. Although this is a true statement, it is not true that just knowing the host can open one of the two unchosen doors to show a goat necessarily means that opening a specific door cannot affect the probability that the car is behind the initially-chosen door. If the car is initially placed behind the doors with equal probability and 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 door are very persuasive, but lead to the correct answer only if the problem is completely symmetrical with respect to both the initial car placement and how the host chooses between two goats ([[Monty Hall problem#refFalk1992|Falk 1992:207,213]]).
Another approach showing switching wins with probability 2/3 is to determine the [[conditional probability]] the car is behind Door 2 given that the player has initially picked Door 1 and the host has opened Door 3 ({{Harvnb|Selvin|1975b}}; [[#refMorganetal1991|Morgan et al. 1991]]; [[#refGrinsteadandSnell2006|Grinstead and Snell 2006:137]]). Referring to the [[decision tree]] as shown to the right ([[#refChun1991|Chun 1991]]) or the equivalent figure below, and considering only cases where the host opens Door 3 after the player picks Door 1, switching loses in a case with probability 1/6 but wins in a case with probability 1/3. The conditional probability the car is behind Door 2 is therefore 2/3 = 1/3 / (1/6 + 1/3), while the conditional probability the car is behind Door 1 is only 1/3 = 1/6 / (1/6 + 1/3).
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==Simpler solution ==
▲An even simpler solution is to reason that switching loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3. ([[#refCarlton2005|Carlton 2005]]).
===Alternative derivations===▼
Formal mathematical derivations can also be given which avoid explicit computations or formula manipulation, and which illustrate various insights into the Monty Hall problem.▼
▲Another way to understand the solution is to consider the two original unchosen doors together. 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 choose the opened door ([[#refAdams1990|Adams 1990]]; [[#refDevlin2003|Devlin 2003]]; [[#refWilliams2004|Williams 2004]]; [[#refStibeletal2008|Stibel et al., 2008]]).
▲As [[Cecil Adams]] puts it ([[#refAdams1990|Adams 1990]]), "Monty is saying in effect: you can keep your one door or you can have the other two doors." The player therefore has the choice of either sticking with the original choice of door, or choosing the sum of the contents of the two other doors, as the 2/3 chance of hiding the car has not been changed by the opening of one of these doors.
▲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.{{' "}}
▲Formal mathematical derivations can also be given which avoid explicit computations or formula manipulation, and which illustrate various insights into the Monty Hall problem.
# '''Proof using a simple solution and symmetry'''. The chance that the door chosen by the contestant, Door 1, hides the car, is 1/3. The conditional probabilities that Door 1 hides the car given that the host opens Door 2, and that he opens Door 3, must be equal to one another, by symmetry. This means that whether or not the host opens Door 3 is [[conditional independence|(statistically) independent]] of whether or not the car is behind Door 1 given the player initially chose Door 1. The conditional probability is equal to the unconditional probability, 2/3. ([[#refGill2011a|Gill, 2011a]]), ([[#refBell1992|Bell (1992)]]: "I will leave it to readers as to whether this equivalence of the conditional and unconditional problems is intuitively obvious.").
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A common variant of the problem, assumed by several academic authors as the [[canonical]] problem, does not make the simplifying assumption that the host must uniformly choose the door to open, but instead that he uses some other [[strategy (game theory)|strategy]]. The confusion as to which formalization is authoritative has led to considerable acrimony, particularly because this variant makes proofs more involved without altering the optimality of the always-switch strategy for the player. In this variant, the player can have different probabilities of winning [[conditional probability|depending on the observed choice]] of the host, but in any case the probability of winning by switching is at least 1/2 (and can be as high as 1), while the [[overall probability]] of winning by switching is still exactly 2/3. The variants are sometimes presented in succession in textbooks and articles intended to teach the basics of [[probability theory]] and [[game theory]]. A considerable number of other generalizations have also been studied.
Some sources, in particular, Morgan et al. ([[#refMorganetal1991|1991]]) state that many popular solutions are incomplete because they do not explicitly address their interpretation of vos Savant's rewording of Whitaker's original question. The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 ''given which door the player has chosen and which door the host opens''.
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According to Morgan et al. ([[#refMorganetal1991|1991]]) "The distinction between the conditional and unconditional situations here seems to confound many." That is, they, and some others, interpret the usual wording of the problem statement as asking about the [[conditional probability]] of winning given which door is opened by the host, as opposed to the overall or unconditional probability. These are mathematically different questions and can have different answers depending on how the host chooses which door to open when the player's initial choice is the car ([[#refMorganetal1991|Morgan et al., 1991]]; [[#refGillman1992|Gillman 1992]]). For example, if the host opens Door 3 whenever possible, then the probability of winning by switching for players initially choosing Door 1 is still 2/3 overall, but only 1/2 if such host opens Door 3, and in contrast 1 if he opens Door 2. In its usual form the problem statement does not specify this detail of the host's behavior, nor make clear whether a conditional or an unconditional answer is required, making the answer that switching wins the car with probability 2/3 equally vague. Many commonly presented solutions address the unconditional probability, ignoring which door was chosen by the player and which door opened by the host; Morgan et al. call these "false solutions" ([[#refMorganetal1991|1991]]). Others, such as Behrends ([[#refBehrends2008|2008]]), conclude that "One must consider the matter with care to see that both analyses are correct."
The version of the Monty Hall problem 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. However, vos Savant made it clear in her second followup column that the intended host's behavior could only be what led to the 2/3 probability she gave as her original answer. "Anything else is a different question" ([[#refvosSavant1991|vos Savant, 1991a]]). "Virtually all of my critics understood the intended scenario. I personally read nearly three thousand letters (out of the many additional thousands that arrived) and found nearly every one insisting simply that because two options remained (or an equivalent error), the chances were even. Very few raised questions about ambiguity, and the letters actually published in the column were not among those few." ([[#refvosSavant1996|vos Savant, 1996]]) The answer follows if the car is placed randomly behind any door, the host must open a door revealing a goat regardless of the player's initial choice and, if two doors are available, chooses which one to open randomly ([[#refMueserandGranberg1999|Mueser and Granberg, 1999]]). The table below shows a variety of ''other'' possible host behaviors and the impact on the success of switching.
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</div>
D. L. Ferguson (1975 in a letter to Selvin cited in {{Harvtxt|Selvin|1975b}}) suggests an ''N'' door generalization of the original problem in which the host opens ''p'' losing doors and then offers the player the opportunity to switch; in this variant switching wins with probability (''N''−1)/[''N''(''N''−''p''−1)]. If the host opens even a single door the player is better off switching, but if the host opens only one door the advantage approaches zero as ''N'' grows large ([[#refGranberg1996|Granberg 1996:188]]). At the other extreme, if the host opens all but one losing door the advantage increases as ''N'' grows large (the probability of winning by switching approaches 1 as ''N'' grows very large).
Bapeswara Rao and Rao ([[#refBapeswaraRao1992|1992]]) suggest a different ''N'' door version where the host opens a losing door different from the player's current pick and gives the player an opportunity to switch after each door is opened until only two doors remain. With four doors the optimal strategy is to pick once and switch only when two doors remain. With ''N'' doors this strategy wins with probability (''N''−1)/''N'' and is asserted to be optimal.
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 formulation is loosely based on [[quantum game theory]]. 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 ([[#refFlitney2002|Flitney and Abbott 2002]], [[#refDArianoetal2002|D'Ariano et al. 2002]]).
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* <span class="citation" id=refBapeswaraRao1992>Bapeswara Rao, V. V. and Rao, M. Bhaskara (1992). "A three-door game show and some of its variants". ''The Mathematical Scientist'' '''17'''(2): 89–94.</span>
* <span class="citation" id=refBarbeau1993>Barbeau, Edward (1993). "Fallacies, Flaws, and Flimflam: The problem of the Car and Goats". ''The College Mathematics Journal'' '''24'''(2): 149-154.</span>
* <span class="citation" id=refBarbeau2000>Barbeau, Edward (2000). ''Mathematical Fallacies, Flaws and Flimflam''. The Mathematical Association of America. {{ISBN
* {{cite book | ref = refBehrends2008 | title = Five-Minute Mathematics | author = Behrends, Ehrhard | publisher = AMS Bookstore | year = 2008 | isbn = 978-0-8218-4348-2 | page = 57 | url = http://books.google.com/?id=EpkyE6JFmkwC&pg=PA48&dq=monty-hall+door-number }}
* <span class="citation" id=refBell1992>Bell, William (1992). "Comment on ''Let's make a deal'' by Morgan et al.", ''American Statistician'' '''46'''(3): 247 (August 1992).</span>
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* {{cite web|ref=refDevlin2003|url=http://www.maa.org/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=2008-04-25}}
* {{cite news | ref = refEconomist1999 | work = The Economist | title = The Monty Hall puzzle | volume = 350 | publisher = The Economist Newspaper | year = 1999 | page = 110 | url = http://books.google.com/books?id=H3vPAAAAIAAJ&q=goat-b+goat-a&dq=goat-b+goat-a&lr=&as_brr=0&as_pt=ALLTYPES&ei=yTLhSbvzJYuIkASxlsinDQ&pgis=1 }}
* <span class="citation" id=refDiaconis1988>Diaconis, Persi (1988). ''Group representations in probability and statistics'', IMS Lecture Notes, Institute of Mathematical Statistics. {{ISBN
* <span class="citation" id=refFalk1992>Falk, Ruma (1992). "A closer look at the probabilities of the notorious three prisoners," ''Cognition'' '''43''': 197–223.</span>
* <span class="citation" id=refFlitney2002>Flitney, Adrian P. and [[Derek Abbott|Abbott, Derek]] (2002). "Quantum version of the Monty Hall problem," ''Physical Review A'', '''65''', Art. No. 062318, 2002.</span>
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* <span class="citation" id=refGardner1959a>[[Martin Gardner|Gardner, Martin]] (1959a). "Mathematical Games" column, ''Scientific American'', October 1959, pp. 180–182. Reprinted in ''The Second Scientific American Book of Mathematical Puzzles and Diversions''.</span>
* <span class="citation" id=refGardner1959b>[[Martin Gardner|Gardner, Martin]] (1959b). "Mathematical Games" column, ''Scientific American'', November 1959, p. 188.</span>
* <span class="citation" id=refGill2002>[[Jeff Gill|Gill, Jeff]] (2002). ''Bayesian Methods'', pp. 8–10. CRC Press. {{ISBN
* <span class="citation" id=refGill2010>[[Richard D. Gill|Gill, Richard]] (2010) Monty Hall problem. pp. 858–863, ''International Encyclopaedia of Statistical Science'', Springer, 2010. Eprint [http://arxiv.org/pdf/1002.3878v2] </span>
* <span class="citation" id=refGill2011>[[Richard D. Gill|Gill, Richard]] (2011) The Monty Hall Problem is not a probability puzzle (it's a challenge in mathematical modelling). ''Statistica Neerlandica'' '''65'''(1) 58-71, February 2011. Eprint [http://arxiv.org/pdf/1002.0651v3] </span>
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*<span class="citation" id=refGill2011b>[[Richard D. Gill|Gill, Richard]] (2011b) Monty Hall Problem (version 5). ''StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies'' 2011. [http://statprob.com/encyclopedia/MontyHallProblem2.html] </span>
* <span class="citation" id=refGillman1992>[[Leonard Gillman|Gillman, Leonard]] (1992). "The Car and the Goats," ''American Mathematical Monthly'' '''99''': 3–7.</span>
* <span class="citation" id=refGranberg1996>Granberg, Donald (1996). "To Switch or Not to Switch". Appendix to vos Savant, Marilyn, ''The Power of Logical Thinking''. St. Martin's Press. {{ISBN
* <span class="citation" id=refGranbergandBrown1995>Granberg, Donald and Brown, Thad A. (1995). "The Monty Hall Dilemma," ''Personality and Social Psychology Bulletin'' '''21'''(7): 711-729.</span>
* {{cite book | ref=refGrinsteadandSnell2006 | author=Grinstead, Charles M. and Snell, J. Laurie | title = Grinstead and Snell’s Introduction to Probability | url=http://www.math.dartmouth.edu/~prob/prob/prob.pdf | accessdate=2008-04-02 | date=2006-07-04 | format=PDF}} Online version of ''Introduction to Probability, 2nd edition'', published by the American Mathematical Society, Copyright (C) 2003 Charles M. Grinstead and J. Laurie Snell.
* {{cite book | ref = refGruber2010 | title = The World's 200 Hardest Brain Teasers| author = Gruber, Gary | publisher = Sourcebooks, Inc. | year = 2010 | isbn = 13:978-1-4022-3857-4 | page = 136 | url = http://books.google.com/books?id=sesXaPoWyb0C&printsec=frontcover&dq=The+World's+200+Hardest+Brainteasers&hl=en&ei=bXzbTaGMIeTUiALCq7kR&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDgQ6AEwAA#v=onepage&q=The%20World's%20200%20Hardest%20Brainteasers&f=false | authorlink = Gary Gruber}}
* <span class="citation" id=refHall1975>[[Monty Hall|Hall, Monty]] (1975). [http://www.letsmakeadeal.com/problem.htm The Monty Hall Problem.] LetsMakeADeal.com. Includes May 12, 1975 letter to Steve Selvin. Retrieved January 15, 2007.</span>
* <span class="citation" id=refHenze1997>Henze, Norbert (1997). ''Stochastik für Einsteiger: Eine Einführung in die faszinierende Welt des Zufalls'', pp. 105, Vieweg Verlag, {{ISBN
* <span class="citation" id=refHerbransonandSchroeder2010>Herbranson, W. T. and Schroeder, J. (2010). "Are birds smarter than mathematicians? Pigeons (''Columba livia'') perform optimally on a version of the Monty Hall Dilemma." ''J. Comp. Psychol.'' '''124'''(1): 1-13. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/20175592 March 1, 2010. http://people.whitman.edu/~herbrawt/HS_JCP_2010.pdf </span>
* <span class="citation" id=refKraussandWang2003>Krauss, Stefan and Wang, X. T. (2003). "The Psychology of the Monty Hall Problem: Discovering Psychological Mechanisms for Solving a Tenacious Brain Teaser," ''Journal of Experimental Psychology: General'' '''132'''(1). Retrieved from http://www.usd.edu/~xtwang/Papers/MontyHallPaper.pdf March 30, 2008.</span>
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* <span class="citation" id=refMueserandGranberg1999>Mueser, Peter R. and Granberg, Donald (May 1999). [http://econpapers.repec.org/paper/wpawuwpex/9906001.htm "The Monty Hall Dilemma Revisited: Understanding the Interaction of Problem Definition and Decision Making"], University of Missouri Working Paper 99-06. Retrieved June 10, 2010.</span>
* <span class="citation" id=refNalebuff1987>[[Barry Nalebuff|Nalebuff, Barry]] (1987). "Puzzles: Choose a Curtain, Duel-ity, Two Point Conversions, and More," ''Journal of Economic Perspectives'' '''1'''(2): 157-163 (Autumn, 1987).</span>
* <span class="citation" id=refRosenhouse2009>Rosenhouse, Jason: ''The Monty Hall Problem''. Oxford University Press 2009, {{ISBN
* {{Cite journal|ref=refRosenthal2005a|title=Monty Hall, Monty Fall, Monty Crawl|first=Jeffrey S.|last=Rosenthal|year=2005a|journal=Math Horizons|pages=September issue, 5–7}} [http://probability.ca/jeff/writing/montyfall.pdf Online reprint, 2008].
* <span class="citation" id=refRosenthal2005b>Rosenthal, Jeffrey S. (2005b): ''Struck by Lightening: the Curious World of Probabilities''. Harper Collings 2005, {{ISBN
* {{Citation |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= |doi= }}
* {{Citation |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= |doi= }}
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* {{cite book | ref = refSchwager1994 | title = The New Market Wizards | author = Schwager, Jack D. | publisher = Harper Collins | year = 1994 | isbn = 978-0-88730-667-9 | page = 397 | url = http://books.google.com/?id=Ezz_gZ-bRzwC&pg=PA397&dq=three-doors+monty-hall }}
* {{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|format=PDF|work=Course notes for Sociology Graduate Statistics I|accessdate=2008-04-25}}
* <span class="citation" id=refWheeler1991>{{cite book | ref = refWheeler1991 | title = Phylogenetic analysis of DNA sequences | chapter = Congruence Among Data Sets: A Bayesian Approach | author = Wheeler, Ward C. | editor = Michael M. Miyamoto and Joel Cracraft | publisher = Oxford University Press US | year = 1991 | isbn = 978-0-19-506698-2 | page = 335 | url = http://books.google.com/?id=1wqvNgz58JQC&pg=PA335&dq=%22monty+hall%22+unchanged+switch }}</span>
* <span class="citation" id=refWhitaker1990>Whitaker, Craig F. (1990). [Formulation by Marilyn vos Savant of question posed in a letter from Craig Whitaker]. "Ask Marilyn" column, ''Parade Magazine'' p. 16 (9 September 1990).</span>
<!-- {{cite journal | author = Marilyn 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 = 2006-11-27 | isbn = 0312081367 }} -->
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{{commons|Monty Hall problem}}
* [http://www.marilynvossavant.com/articles/gameshow.html The Game Show Problem]–the original question and responses on Marilyn vos Savant's web site
* {{tlp|dmoz|Science/Math/Recreations/Famous_Problems/Monty_Hall/|Monty Hall}}
* "[http://demonstrations.wolfram.com/MontyHallParadox/ Monty Hall Paradox]" by Matthew R. McDougal, [[The Wolfram Demonstrations Project]] (simulation)
* [http://www.nytimes.com/2008/04/08/science/08monty.html The Monty Hall Problem] at The New York Times (simulation)
* [http://dl.dropbox.com/u/548740/www/DoorKeeperGame/index.html The Door Keeper Game] (simulation)
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