Discuss The Monty Hall Problem


Well-Known Member
4 Jun 2011
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 #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?


Is it to your advantage to switch your choice?
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Active Member
11 Jun 2011
This question, named "the Monty Hall problem" because of its similarity to scenarios on the game show Let's Make a Deal, existed long before being posed to Marilyn vos Savant (whose IQ of 228 is the highest ever recorded), but was brought to nationwide attention by her "Ask Marilyn" column in Parade magazine in 1990.

Vos Savant answered arguing that the selection should be switched to door #2 because it has a 2/3 chance of success, while door #1 has just 1/3. Or to summarise, 2/3 of the time the opened door #3 will indicate the location of door with the car (the door you hadn't picked and the one not opened by the host). Only 1/3 of the time will the opened door #3 mislead you into changing from the winning door to a losing door. These probabilities assume you change your choice each time door #3 is opened, and that the host always opens a door with a goat. This response provoked letters of thousands of readers, nearly all arguing doors #1 and #2 each have an equal chance of success. A follow-up column reaffirming her position served only to intensify the debate and soon became a feature article on the front page of The New York Times. Among the ranks of dissenting arguments were hundreds of academics and mathematicians.

Under the most common interpretation of the problem where the host opens a losing door and offers a switch, vos Savant's answer is correct because her interpretation assumes the host will always avoid the door with the prize. However, having the host opening a door at random, or offering a switch only if the initial choice is correct, is a completely different problem, and is not the question for which she provided a solution. Vos Savant addressed these issues by writing the following in Parade Magazine, "...the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. Anything else is a different question." In vos Savant's second followup, she went further into an explanation of her assumptions and reasoning, and called on school teachers to present the problem to each of their classrooms. In her final column on the problem, she announced the results of more than a thousand school experiments. Nearly 100% of the results concluded that it pays to switch. Of the readers who wrote computer simulations of the problem, 97% reached the same conclusion. A majority of respondents now agree with her original solution, with half of the published letters declaring the letter writers had changed their minds.