June 6, 2008
To: Brian Hayes
c/o: American Scientist
I read with a great deal of interest your column in the American Scientist, July-August 2008, entitled Programs and Probabilities, in which you discuss the Monty Hall affair. This was discussed in the New York Times by John Tierney on April 8, 2008 and at that time I wondered why such a simple problem could generate so much controversy.
Your Monte Carlo results suggest that switching doors doubles your chances of winning but does not really explain why that is happening. The probabilistic explanation offered by "the other side" in your article is correct but does not explain in simple terms why that result occurs.
What if one wanted to explain what is happening to a bright high school sophomore who has never studied probability? There is a direct approach which is similar to what happens in our thinking in a chess game: if I do this then this happens but if I do this other thing then that could happen. And the problem here is much simpler than any real chess game.
Here's the direct approach:
Let's assume without any loss of generality that my initial choice is always door 1. There are three different cases to consider: the prize itself is behind door 1 or it is behind door 2 or it is behind door 3.
Let's now assume the prize is behind door 1. Monty Hall opens either door 2 or door 3. In either case I have two choices: I can stay with my original choice, door 1, or I can switch to the door that is still closed (to door 2 if Monty opens door 3 or to door 3 if Monty opens door 2). If I don't switch I win the prize and if I switch I lose.
Now assume the prize is behind door 2. Monty opens door 3 and shows me that it is empty. If I don't switch (stay with door 1) I lose but if I switch to door 2 I win.
Finally assume the prize is behind door 3. Monty opens door 2 and shows me that it is empty. If I don't switch (stay with door 1) I lose but if I switch to door 3 I win.
Thus if I don't switch I win only when the prize is behind the door of my original choice (door 1) but if I switch I win in two cases, when the prize is behind door 2 or is behind door 3.
The switching strategy does twice as well as the "don't switch" strategy.
Of course this is closely related to the probabilistic approach but if the problem is approached in this direct way I don't think there would have been any controversy.
One more thought: sticking with the original choice neglects the additional information given to us by Monty when he opens a door. The additional information given to us by Monty when he opens door 3 is that he didn't open door 2. The additional information given to us by Monty when he opens door 2 is that he didn't open door 3. The switching strategy takes advantage of this additional information and may help the reader understand that there is no paradox involved here.
Los Angeles, CA
posted by Ethan Aronoff
July 15, 2008