# Monty Hall Problem aka: Game Show Problem

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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 opening 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 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 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 surprising amount of Internet Backdraft. 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, see the Real Life folder for how 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.

## Examples:

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Card Games

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%.

Literature

Live-Action TV
• Let's Make a Deal is the Trope Namer for the most common formulation of the problem, as mentioned above.
• 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.
• 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, but Cochran did end up being Right for the Wrong Reasons — the unseen item really was better.
• 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 arguing over it; Holt still believes that switching is pointless and the odds are 50/50, so he ends up furiously arguing with Kevin, who's presenting the correct (2/3) answer. Santiago thinks the best way of getting them to make up is to try and get Holt to understand the answer, while Diaz, armed with the knowledge that Holt's been working the night shift a lot recently and hasn't had much free time, tells Holt to his face that he's just pent-up and "You two just need to bone."

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'.
• Referenced in Zero Time Dilemma, where an entire fragment is named for this problem, including demonstrating the problem in a slightly modified form. In the relevant room, there are 10 lockers. Only one locker has a gas mask. After a selection is made, 8 of the lockers open, all of them empty. Then, the player is asked if they want to stick with their original choice, or to switch to the other locker. The problem is discussed by the characters during this scenario.
• The puzzle is presented in Umineko: When They Cry as a way to encourage Ange to hold onto her version of what happened all those years ago.

Web Comics
• 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, 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
• Marilyn Vos Savant, author of Parade magazine's "Ask Marilyn", infamously said that the correct answer was to switch. (She addressed the ambiguities in a follow-up column.)
• 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 that Savant 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".
• Some people (including Vos Savant herself) have proposed variants of the problem to explain why the answer is right. One has 1 million doors instead of 3, the person picks a random door (say 472,865) and the host open all doors except, say 983,724, and ask if you want to switch (don't worry about how he'd open 999,998 doors on his own). This makes it more obvious, as is only 1 in a million at first, so obviously after he removes all but one wrong door, the one left behind after would be more likely to be right.

"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

### Alternative Title(s):Game Show Problem

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