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Monty Hall Problem

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"If the host is required to open a door all the time and offer you a switch, then you should take the switch, But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood.
My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go home."
Monty Hall, July 1991

A classic mathematical problem involving probabilities. 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 any one of three doors to claim whatever is behind the chosen door. One door 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 other two doors, revealing a goat. The contestant is then offered the choice to switch to the unrevealed door or stick with their 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 switching will always get you the opposite of what you initially picked. 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 surprisingly large Flame War cascade when mentioned. Even the initial article in 1990 by Marilyn Vos Savant got a pre-internet version of this: a lot of people (including a lot of math professors) wrote in saying she was wrong and it didn't matter if you switched or not. It turned out she was correct after all, as demonstrated by this table.

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. If he does pick at random then you do gain new information when a goat is revealed, because it might not have happened; the chances are then 50% that the car is behind each remaining door. One reason why many people give the wrong answer to the original problem is subconsciously confusing it with this version. Especially egregious examples may not include the notion that the host will always give the player the opportunity to switch at all, allowing for the possibility that the host only allows you to switch if you pick the car (in which case switching gets you a 100% chance of getting a goat).

How it worked out on the show is irrelevant, especially since Monty Hall himself in this interview denies that he ever actually did this deal. That said, he tested it out in 1991 and showed that while the Monty Hall Problem applies to many things, it didn't apply to Monty Hall.

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


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    Fan Works 

    Films — Live-Action 
  • 21, where they manage to completely screw up the answer to the problem: the teacher presenting the problem specifically states that Monty (who knows what's behind each door) may be using "reverse psychology" in an attempt to trick him. The student says this doesn't matter - but it does, because if it is indeed a trick and Monty only offers the switch when the correct door is picked, then the odds of switching getting you the car becomes 0%, not 66.6%.

  • The Curious Incident of the Dog in the Night-time: The protagonist explains the Monty Hall problem to the reader, using a tree diagram to illustrate all the different ways it can play out and the odds of each possible outcome.
    So if you change, 2 times out of 3 you get a car. And if you stick, you only get a car 1 time out of 3.
    And this shows that intuition can sometimes get things wrong. And intuition is what people use in life to make decisions. But logic can help you work out the right answer.
    It also shows that Mr. Jeavons was wrong and numbers are sometimes very complicated and not very straightforward at all. And that is why I like The Monty Hall Problem.
  • The Monty Hall problem is explained six different ways (including a list of everything that might happen) in Ian Stewart's The Magical Maze.
  • In Conned Again, Watson! by Colin Bruce, a book explaining probability theory via Sherlock Holmes stories, the mathematician Charles Dodgson sets up a Monty Hall Problem scenario for Watson to pass the time on a train journey. Watson insists the odds are fifty/fifty until Dodgson suggests they switch places. Once he sees how his options in revealing an empty box are constrained by which box Dodgson picked, he gets it.

    Live-Action TV 
  • Let's Make a Deal is the Trope Namer for the most common formulation of the problem, as mentioned above, but rarely if ever allowed it to play out verbatim. The show usually offered a lump sum of cash instead of the switch.
  • Mentioned but averted 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, so the fact 24 of the 26 cases had been opened did not affect the values of the two remaining cases.
  • 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.
  • Survivor:
    • Incorrectly invoked in the Caramoan season. 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, but Cochran did end up being Right for the Wrong Reasons — the unseen item really was better.
    • At the final seven of Survivor 41, the first contestant who dropped out of the Immunity Challenge had to play this game, where the "car" is safety and the "goats" are elimination. Deshawn chose not to switch, and despite the odds being against him, this choice was correct.
  • Discussed in NUMB3RS, as most mathematical concepts are. It turned out to be an example of Chekhov's Classroom, although in this case teaching the Monty Hall Problem is what helped Charlie have a "Eureka!" Moment.
  • The Price Is Right had a pricing game called Barker's Marker$ which imposed a four-way Monty Hall problem. The game board had four prices, three of which matched prizes on display. The contestant marked three prices and, after two were revealed, had the option of switching the last marker to the other price at a cost of $500 given to the contestant at the start of the game. The decision brings the problem into play where the contestant, after blindly picking three prizes, has a 75% chance of winning if the choice is made to switch.
  • Brooklyn Nine-Nine plays this for comedy by having Captain Holt and his husband Kevin get into a running argument over it. Santiago desperately tries to resolve the argument by trying to convince Holt of the right answer. Diaz, on the other hand, points that they're only arguing because they haven't been spending much time together and they're both testy. She tells Holt to his face "You two just need to bone." Guess who's solution works.
  • The old game show ''Play Your Hunch" (1958-63) employed basic intuition into determining which of three objects was the answer to a question.

    Video Games 
  • Implemented 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'.

    Visual Novels 
  • The puzzle is presented using slices of cakenote  in Umineko: When They Cry as a way to encourage Ange to hold onto her version of what happened all those years ago (by way of metaphor, all of the alternate theories here are slices that "can't possibly hold the prize", so instead switching is a bad idea).
  • Referenced in Zero Time Dilemma, where an entire fragment is named for this problem, including demonstrating the problem in a slightly modified form. In a room filling with Deadly Gas, there are 10 lockers, only one of which has a gas mask in it. After you choose a locker, the game opens 8 of the empty ones (not including the one you picked, obviously) and asks you to choose again from the remaining 2 lockers. The problem is discussed by the characters during this scenario. Interestingly, the fact that choosing the other locker merely increases your chance of winning the prize is taken into account, as it's entirely possible for you make the "correct" choice and still wind up with an empty locker.

  • Parodied in this xkcd. The Existentialist in a Beret delightedly walks off with the first goat revealed, instead of making the choice., because he doesn't consider getting a goat to be a failure state. 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, the villains 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!
  • The cat in the Looney Tunes cartoon "Early To Bet" is in crutches by the end of the film after enduring punishments from losing at gin rummy to a bulldog. When the bulldog finally calls the game off, not wanting the cat to suffer any further, the Gambling Bug (which caused the cat to play gin rummy with the dog in the first place) offers to play the cat for high card. The cat draws a three. "Not so good, cat. Watch." The Gambling bug's draw: a two. The cat dishes out his own punishment to the Gambling Bug—a rolled up copy of the Post ("No, no! Not the Post!")
  • Subverted in The Hollow. When put into an impromptu game show where the group has to choose one of three doors to find a portal leading forward, Mera assumes that the situation is one of these and wants them to switch after the host reveals a mule behind one of the doors. The others are confused by her logic, but ultimately decide to trust her anyway. She's wrong and they end up having to fight a monster who they must trick into breaking open the portal door instead.

    Real Life 
  • In July 1991, Monty himself was asked to help settle the debate once and for all, with the tests done in his home with a set of keys representing the car and a couple of cheap snacks representing the goats.
    • Monty started out with doing the problem as commonly stated, eventually proving Marilyn Vos Savant's statement that switching was correct. After this, he read her original article and noticed something she wasn't considering...
    • The next set of tests saw Monty in full-on Big Dealer mode, which pretty much meant all bets were off: he'd sometimes open the chosen door immediately, or offer various amounts of cash to call off the deal in lieu of a switch. As Monty noted, "there's the psychological factor to consider".

Alternative Title(s): Game Show Problem