The Game-Show Doors Question Revisited
Charles Fishman of Ridgewood, N.J., writes:
Marilyn: You answered a question about a person changing his or her selection of one door after another of three doors is opened and a booby prize is revealed [read article]. You wrote that it does not help to switch doors.
Yet I recall your previous columns, starting back in 1990, about a similar question. You wrote that you increase your chance of winning if you switch doors after being shown a losing door. This caused quite an uproar from the public and academia, who insisted there was no advantage to switching from the door originally chosen. As you could not seem to convince the naysayers after several tries, you asked the nation’s schools to try an experiment and report the results. They did, and you were proved correct.
Could you please explain your recent, seemingly contradictory answer? I am guessing there is a subtlety that I am missing.
Charles: After I wrote about this question extensively some years ago, my answer and explanations began to appear in dozens of textbooks. Since then, an unexpected turn of events has occurred.
Back in 1990, everyone was convinced that it didn’t help to switch, whether the host opened a losing door on purpose or not. Assuming a knowledgeable host who would always open a losing door, that was incorrect. (A knowledgeable host who opened a winning door on purpose wouldn’t have much of a show, would he?!)
Now everyone is convinced that it always helps to switch, regardless of what the host knows. But this is just as incorrect!
Everything depends on what the host knows. If he knows what’s behind all the doors and always opens a losing door on purpose, your chances go up if you switch. But if he doesn’t know what’s behind the doors, and he opens a losing door by chance, your chances do not go up if you switch doors.
In the new question, the host is supposed to be knowledgeable, but he forgets which door hides the car. It doesn’t matter that you don’t know this. The question is still, “Is it to your advantage to switch?” The answer is “no.”
Here’s one way to look at it. A third of the time, the clueless host will choose the door with the prize, and the game will be over immediately. In our puzzle, that didn’t occur. So we’re considering the two-thirds of the time when either: 1) You have chosen the door with the prize; or 2) The prize is behind the unopened door. Each of these two events will occur one-third of the time, so you don’t gain by switching.
Following is my favorite letter on the subject.
Liz Bancevich of Newaygo, Mich., writes:
Marilyn: You were wrong about the puzzle with the car and the goats. As far as I am concerned, the goats are the prizes! I know your reader wanted to include the most disgusting prize in the world for this puzzle, but next time, please change the wording to “400 pounds of raw liver” or something like that. Goat lovers are offended! In a few hours, I’ll be going out in the rain to feed my herd of thirteen goats. It is a pleasure to be with them even when the temperature is cold, and they are a bit cranky.
Answer: A. Vicar. Despite the number of poems he wrote about imaginary mistresses, Herrick led a relatively solitary life in the English countryside as an Anglican clergyman.
I believe that all of the posts so far, including my previous one, have overlooked something in Marilyn's solution: "A third of the time, the clueless host will choose the door with the prize, and the game will be over immediately." But what happens then? If the player is given the prize, that would be an assumption that the player would change his selection. Without knowing the actual rules of the game, there is no point to this discussion.
When the "3-doors" puzzle first appeared I saw that the "2 out of 3" answer was correct, but I viewed the puzzle from a logical standpoint, not a mathematical one. The way i see it, the host has 3 somethings (doors, boxes?) of equal value (a 1 in 3 chance of winning something). The contestant takes one, leaving the MC with the remaining two. When the MC opens one of his and offers the contestant a chance to switch, the result from the contestant's viewpoint is exactly the same as if the MC had not opened one and instead had offered to trade his two "somethings" for the contestant's one. Looking at the new version this way, the answer is the same. It does not matter what the MC knows, or how he manages to find a losing door to expose. In the end, he is still offering a 2 for 1 trade.
Re: doktorkev If the contestant does not know whether or not the host knows where the prize is, then it always pays to switch in the sense that he will win either 2/3 of the time or 1/3 of the time depending on whether the host knows or doesn't know where the prize is.