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Monty Hall Problem
aka: Game Show Problem
A classic mathematical problem involving probabilites. The basic form is based on one of the games on the Game Show Lets Make A Deal. The contestant is offered the choice of three doors. One has a car behind it, the two others hide 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 The Other Wiki 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 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 Internet Backdraft.

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 here denies that he ever actually did this).

Not to be confused with Monty Haul, which is a different problem altogether.

Examples of this in works:

Card Games

Film

Literature

Live-Action TV
  • Lets Make A Deal is the Trope Maker and Trope Namer for the most common formulation of the problem, as mentioned above.
  • Subverted in Deal or No Deal. 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
  • 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.)
  • James May's Man Lab did a Russian Roulette version of this with beer cans called, wait for it, "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 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.

Video Games
  • Implemented in in Sandcastle Builder, 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 ''xkcd' 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'.

Webcomics
  • Parodied in xkcd. The Existentialist in a Beret delightedly walks off with the first goat revealed, instead of making the choice. According to the Alt Text, the other goat drove off in the car a few minutes later.

Western Animation
  • In the first episode of The 13 Ghosts of Scooby-Doo, Those Two Bad Guys 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 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 dumb hunch.
    Weird: Well, let's see just how dumb your hunch really was!

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.)

Knights and KnavesStock PuzzleRiddle Me This
'Mission: Impossible" Cable DropShout-Outs IndexNighthawks Shot
Mistaken for JunkieJustForFun/Tropes Examined by the Myth BustersNo Sense of Direction

alternative title(s): Game Show Problem
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