The Monty Hall Problem

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 other 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 take the switch?

That is the way the “Monty Hall Problem” is usually stated. Marilyn vos Savant published the problem with essentially this wording in 1990. In the recent film “21″ the problem appears in this form. Unfortunately, when stated this way, there is no indication in the problem whether the the host always must open a door with a goat or not. That condition is crucial. The answer to the problem is that if the host must always open a door with a goat, it is to the player’s advantage to switch — there is a 2/3 chance that switching will win. However, if the problem does not state whether or not the host must open a door, the problem is indeterminate. The Wikipedia now has a good explanation of this answer so I won’t go into it in detail here. But consider this: suppose the host only opens a door to show a goat if you have chosen the correct door with the car? In that case, switching will lose 100% of the time. That is why the requirement that the host must always open a door is important.

I can’t believe people are still leaving this out! The problem is confusing enough without making it indeterminate. Even veteran science writer Brian Hayes in the prestigious magazine American Scientist got the statement of the problem wrong (twice!) although his explanation of the solution in the article was correct.

In the Wikipedia article you will also find a much better statement of the problem:

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. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you “Do you want to switch to Door Number 2?” Is it to your advantage to change your choice?

So, please excuse my yelling, but if you are posing this problem for somebody, or just discussing it, make sure you specify that the host must open a door! Nothing substitutes for precision in stating a problem. Otherwise you might just have a Bad Question.