This issue comes across as a bad joke that keeps getting repeated. Someone asked Marilyn Vos Savant a question several years ago, and still today I get e-mail challenging whether she was right. Of course, I get it because I wrote an analysis of the problem as it was recounted to me (not having seen the original statement of the problem, I admittedly was relying on hearsay), and stated that her analysis was incorrect. Those who agree or disagree with her have written; I interacted with two of those on a second page on the subject. Still they continue to write.
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George Peltz wrote recently, and claimed that Marilyn was correct and I was mistaken. I get that often. He maintained that a simple mathematical analysis of the problem provides her answer. We'll consider that in a moment. The reason I'm mentioning Mr. Peltz, and writing yet a third page on this subject, is that Mr. Peltz has sent me a copy of what he states is the original statement of the question, which appears to contain the elements claimed by most of those who have seen it, and gives a general impression of authenticity. Finally, assuming this is the original statement, we can look at how the problem was worded, and see where I, or Marilyn, made our mistakes.
Here is the problem, as he presents it:
| 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 says to you, 'Do you want to pick door No. 2?'
Is it to your advantage to switch your choice of doors? |
Now that I have seen the problem, I also see why the solutions offered by Mr. Major and Mr. Schipsi (cited on an earlier page) are so different. Within the question there is a failure to distinguish the rules of the game from the actions of the host. It appears on careful examination that these, and only these, are stated as rules:
Everything else that is included in the statement of the problem is a description of the actions of the host, not a rule of the game. Therein lies the confusion.
Perhaps I can clarify the issues by presenting all the possiblities.
The question stipulates that there are three doors, and that you picked #1; it also stipulates that one of those doors hides a car and the other two hide goats. At this moment there are three possibilities:
What is not stipulated is that the game cannot end here. That means we double the possibilities thus:
Indeed you do have a 1/3 chance that you have the car, and there is a 2/3 chance that you do not; however, the host is about to open door number #3, and reveal that it does not have the car. This eliminates certain possibilities, because the game has not ended; however, some of those possibilities must be considered still.
That is, the possibilities
and
The game does not stipulate that that the host must offer you another door; only that he does. This means that once the host offers to trade doors the possibilities are:
and
This has nothing to do with the host of a game show. It has to do with the fact that "rules of the game" are being assumed but not stated, specifically, that the host must continue the game after you have chosen. That is nowhere stated in the question; it is presumed in Marilyn Vos Savant's answer, and without that presumption her answer is incorrect.
You cannot answer this as a strictly mathematical model, because as a mathematical model this is the question:
| 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. What is the probability that the host will take any action at all thereafter? Given that the host does take action, what action will he take? If he knows you have the wrong door, will he end the game? If he knows you have the right door, will he attempt to persuade you to trade it for a wrong door? |
The rest of what the question says, about what the host does, is not part of the mathematical probabilities of play but part of the psychology of game strategy. You cannot conclude that because you only had a 33% chance of having the right door initially that there is now a 67% chance that he is offering you the right door; to reach that, you must first know that he would offer you the right door if you had the wrong door.
Consider it from the other end.
Let us suppose the host is secretly benevolent, and wants to give you the car. You choose door number one. If he wants you to win the car, how will he act? If the car is behind door number one, he will immediately say, "Congratulations, you won a car!" On the other hand, if the car is behind door number two, he will say, "Well, look, door number three is wrong; but wouldn't you rather trade whatever it is that's behind door number one for whatever is behind door number two?" In the case of the benevolent host, the answer is, Absolutely. You have a 100% chance of getting the car by taking the door he suggests, and a 0% chance of having it already.
In exactly the same way, if the host is playing to beat you, the moment you have picked door number one he will say, "I'm sorry, you lose, that's a goat, the car was behind door number two;" or, if it is behind door number one, he'll say, "Well, look behind door number three, see, there's a goat; now there could be a goat behind door number one, would you like to swap that for door number two?"
The problem with a "strictly mathematical" analysis is that it ignores the fact that the problem does not state that showing another door or offering to swap doors is required in the rules of the game; it only states that this is what the host does in this particular instance. The "probability" that you will do better by taking the door is not based on your odds of having the right or wrong one initially, but entirely on the game objectives and strategic options of the host, which are not in any way restricted in the question.
I hope that clarifies the matter.
My original consideration of the problem is posted here, and the presentations of Mr. Major and Mr. Schipsi, and my responses to those comments, here.