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One of the most vexing paradoxes in probability theory is the Monty Hall problem, which is from a 1960s TV show called "Let's Make a Deal", hosted by Monty Hall. The standard setup is that the player is shown 3 doors. Only one has a valuable prize.

After the player picks a door, Monty Hall takes the two unpicked doors (2/3 chance of it being there) and opens up one of them, never the one with the prize. Now there are only 2 doors. He then offers the player a chance to switch.

The puzzle has been very confusing. When Marilyn vos Savant proposed a solution, as many as 1000 people with doctorates challenged her, some challenging her results, others her presumptions about specific game host behaviour. Yet she stuck to her guns, and made her point to many. However, why is there so much debate about theory and rule meanings, when a canonical source exists to answer both?

The Monty Hall problem is, after all, modeled on a real thing with a statistically significant number of plays.

My question is not about the mathematical theory, and the Internet is already heavy with that debate; please add none here. This quesion is: has anyone ever gone back and looked at actual tapes, films or business records of the Let's Make a Deal show, and tallied how the contestants actually did?

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    Part of the problem with this question is that many believe (and I'm not taking sides here, just pointing it out), that the problem as stated does not actually follow how the show actually works. If that is true, then it doesn't make sense to go back and check how they did because the show doesn't work that way. – user1118321 Jan 28 at 3:32
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    This doesn't look like "Trivia questions that do not add to the understanding or appreciation of a movie/TV-show" Its an analysis question – KharoBangdo Jan 28 at 7:18
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    Note that there is no controversy. There is a right answer, which can be proven easily enough, and many wrong answers, so I don’t understand what this exercise is meant to accomplish. – James McLeod Jan 28 at 12:44
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    I wasn't insinuating a discussion on the mathematical validity of the problem itself, but...if the problem doesn't actually follow how the show works (at least as posited in that comment), that very much is some kind of answer to this question, if reasoned properly, of course. – Napoleon Wilson Jan 28 at 16:20
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    @NapoleonWilson oh, you're right, duh, I knew that... well, thank you, to whoever was there behind door #3... – Harper Jan 29 at 17:08
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As @user1118321 has pointed out no one has done this because the game is not played the way it is described in the Monty Hall Problem.

In 1991 former Let's Make a Deal host Monty Hall created a simulation of the show for NY Times reporter John Tierney and found the flaw in trying to use the mathematical problem to play the actual game. From the NY Times article Behind Monty Hall's Doors: Puzzle, Debate and Answer?

After the 20 trials at the dining room table, the problem also captured Mr. Hall's imagination. He picked up a copy of Ms. vos Savant's original column, read it carefully, saw a loophole and then suggested more trials.

On the first, the contestant picked Door 1.

"That's too bad," Mr. Hall said, opening Door 1. "You've won a goat."

"But you didn't open another door yet or give me a chance to switch."

"Where does it say I have to let you switch every time? I'm the master of the show. Here, try it again."

On the second trial, the contestant again picked Door 1. Mr. Hall opened Door 3, revealing a goat. The contestant was about to switch to Door 2 when Mr. Hall pulled out a roll of bills.

Monty then offered the contestant increasing amounts of money, stopping at $5,000 to not switch doors. The contestant refused the money and switched doors.

"You just ended up with a goat," he said, opening the door. The Problem With the Problem.

Mr. Hall continued: "Now do you see what happened there? The higher I got, the more you thought the car was behind Door 2. I wanted to con you into switching there, because I knew the car was behind 1. That's the kind of thing I can do when I'm in control of the game. You may think you have probability going for you when you follow the answer in her column, but there's the psychological factor to consider."

He proceeded to prove his case by winning the next eight rounds. Whenever the contestant began with the wrong door, Mr. Hall promptly opened it and awarded the goat; whenever the contestant started out with the right door, Mr. Hall allowed him to switch doors and get another goat. The only way to win a car would have been to disregard Ms. vos Savant's advice and stick with the original door.

And here is why you can't verify the Monty Hall Problem by looking at the game show:

Was Mr. Hall cheating? Not according to the rules of the show, because he did have the option of not offering the switch, and he usually did not offer it.

So basically it's as @user 1118321 said, the game on the show is not played according to the rules of the mathematical problem.

  • That, too, would be provable; the frequency with which Monty simply awards the goat would approach 2/3. – Harper Jan 29 at 16:38
  • @Harper True, but it wouldn't verify the Monty Hall problem which states that the host MUST offer the option to switch. – Legion600 Jan 29 at 17:19
  • It would be exposed as an academic fiction, and force a correct statement of the problem. No wonder so many people struggle with the problem, since circa 1990 most would have actually seen the show regularly. – Harper Jan 29 at 17:24
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    @Harper Even with a correct statement of the problem people still don't understand the solution or why switching is the right decision. It's a matter of people believing their intuition instead of math. – Legion600 Jan 29 at 17:39
  • According to wikipedia, en.wikipedia.org/wiki/Monty_Hall_problem , Marilyn vos Savant's original wording was: <<Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?>> There's nothing in there that says the host has to switch. – Acccumulation Jan 29 at 23:23

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