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A classic mathematical problem involving probabilites. The basic form is based on one of the games on the Game Show Let's 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.
Not to be confused with Monty Haul, which is a different problem altogether.
- 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 The Magical Maze.
- 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.
- 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.)
- 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.)