A classic mathematical problem involving probabilites. The basic form is based on one of the games on the GameShow ''Series/LetsMakeADeal''. The contestant is offered the choice of three doors. One has a car behind it, the two others hide [[{{Zonk}} goats]]. The contestant chooses a door. The host (who knows what is behind each door) then opens one of the two other doors, revealing a goat. The contestant is then offered the choice to switch to the unrevealed door or stick with his original decision. The correct answer is to switch, as the probability is 66.7% that the car will be behind the other door. This is because there was a 2 in 3 chance that you chose a goat originally, and the host isn't providing any new information since he can ''always'' open a door with a goat. See TheOtherWiki [[http://en.wikipedia.org/wiki/Monty_Hall_problem for an explanation of the math]]. Note that this number is true only if the host is ''required'' to reveal a goat and then offer the contestant the choice to switch. See ''[[http://www.nytimes.com/1991/07/21/us/behind-monty-hall-s-doors-puzzle-debate-and-answer.html?pagewanted=all The New York Times]]'' for what happens when the host is not.

Named after the longtime host of ''Let's Make a Deal''. It causes a surprising amount of InternetBackdraft.

This problem is often presented with a flaw where the question does not include the notion that the host will always reveal a goat, as opposed to revealing either of the unpicked doors at random. In the latter case, your odds do not improve one way or the other, even if the car remains unrevealed.

How it worked out on the show is irrelevant, especially since Monty Hall himself is alleged to have said that he usually offered the switch only if the contestant had picked correctly in the first place (and in an interview [[http://www.youtube.com/watch?v=c1BSkquWkDo here]] denies that he ''ever'' actually did this).

Not to be confused with MontyHaul, which is a different problem altogether.
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!!Examples of this in works:

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[[folder: Card Games ]]

* ''PerplexCity''

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[[folder: Film ]]

* ''[[Film/TwentyOne 21]]'', where they manage to completely screw up the answer to the problem. The student says that it doesn't matter if Monty only offers the switch when you pick the correct door, when in fact, if Monty only offers the switch when you pick the correct door, switching gives you a 100% chance of receiving a goat.

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[[folder: Literature ]]

* ''[[Literature/TheCuriousIncidentOfTheDogInTheNightTime The Curious Incident of the Dog in the Night-Time]]''
* Explained six different ways (including a list of everything that might happen) in Ian Stewart's ''TheMagicalMaze''.

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[[folder: Live-Action TV ]]

* ''Series/LetsMakeADeal'' is the TropeMaker and TropeNamer for the most common formulation of the problem, as mentioned above.
* Subverted in ''DealOrNoDeal''. While a contestant who reached the final case was offered the opportunity to switch it out with his/her case, Howie Mandel went out of his way to explain that this was ''not'' a Monty Hall situation: The show offered the switch to everyone who got that far, and he had no knowledge of which case contained which dollar amount.
* ''{{Numb3rs}}''
* ''Series/MythBusters'' not only tested the probabilities of the Monty Hall problem as stated above, but also contestant behavior when presented with the situation. (All 20 "contestants" tested stuck with their original decision rather than switching.)
* ''Series/JamesMaysManLab'' did a RussianRoulette version of this with beer cans called, wait for it, "[[TheDeerHunter The Beer Hunter]]." The rules were simple: there would be three cans, two of which were shaken up. James would pick one can, but always change his mind after Tom took away a "dangerous" can, and Simmy would be left with the one that James originally picked. They would then hold the cans next to their face and open the cans together. They did this for one hundred rounds; along with getting hypothermia and minor carbon dioxide poisoning, James also proved this version true by winning 40:60.
* Incorrectly invoked in ''Series/{{Survivor}}: Caramoan''. When Reynold is given a choice between the slice of pizza he has already won or an unseen item, Cochran tells him that this is the Monty Hall Problem and Reynold should pick the unseen item. This is not the Monty Hall problem at all, although Cochran did end up being right for the wrong reasons because the unseen item WAS better.

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[[folder: Video Games ]]

* Implemented in in ''Videogame/SandcastleBuilder'', and can be played multiple times. Rather than a car, the prize for picking the correct door is gaining 50% of your sandcastle balance. You lose all your sandcastles if you choose incorrectly, but receive a goat as a consolation, as this is largely a reference to the ''Webcomic/{{xkcd}}' [[ParodiedTrope parody]]. Unlike that comic, if a goat is revealed behind a door you didn't pick (which does not always occur, in order to make it harder to figure out whether or not to take a switch when offered), you can't choose to keep it: if you want a goat you have to find the other goat. In order to sow confusion, this game feature is named 'Monty Haul Problem'.

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[[folder: Webcomics ]]

* [[ParodiedTrope Parodied]] in ''Webcomic/{{xkcd}}''. The Existentialist in a Beret delightedly walks off with the first goat revealed, instead of making the choice. According to the AltText, the other goat drove off in the car a few minutes later.

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[[folder: Western Animation ]]

* In the first episode of ''WesternAnimation/The13GhostsOfScoobyDoo'', ThoseTwoBadGuys trick Scooby and Shaggy into opening the Chest of Demons this way. Posing as a game show (In an ancient castle in the Himalayas, no less!) Weird offers the two the following prizes; the [[CoolPlane Mystery Flying Machine]], a deluxe dog house, or they can take whatever's in the box. It didn't end well.
-->'''Weird:''' So, you took the box. Why?
-->'''Shaggy:''' Just call it a [[{{Irony}} dumb]] hunch.
-->'''Weird:''' Well, let's see just how [[{{Foreshadowing}} dumb]] your hunch really was!

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[[folder: Real Life ]]

* Marilyn Vos Savant, author of ''Parade'' magazine's Ask Marilyn, is one of the proud few who got it completely right. (She addressed the ambiguities in a follow-up column.)
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