Edwin, I think that this is a far trickier problem than it appears to be and, in fact, I got the wrong answer when I tried it (I had worked this years ago, but couldn’t remember a thing other than it paid to switch), but for the life of me couldn’t figure out what I was doing wrong after attacking it in the simplest way I knew how (there are other surer but more complicated approaches). I finally had to e-mail my number one son, the real mathematician in the family, to tell me where I had gone wrong.

I had counted 4 possible outcomes: 1) The original pick being the car and the game host picks goat 1. 2) The original pick is the car and the game host picks goat 2. 3) The original pick is goat 1 and the game host shows goat 2. 4) The original pick is goat 2 and the game host shows goat 1.

I took the above to be 4 equally probable outcomes, two of which (1 & 2) would cause a loss if the player switched and two of which (3 & 4) would cause a win if the player didn’t switch. So the probability was 2/4 in each case, so there was no advantage to switching.

Why the above is wrong is very subtle (at least to me whose long suit in mathematics was never probability, although I’m not exactly a total slouch). I was counting the number of possible outcomes, after both actions ((i) initial choice, (ii) revealing a door) have taken place. And of these there are correctly 4. But the error here is that the probability of winning-after-switching should be calculated based on how many initial choices the player has, not how many distinct ways the game can end. And I don’t think that this is all that obvious to see, but I do now. (see Wikipedia’s first solution at

http://en.wikipedia.org/wiki/Monty_Hall_problem).

Well, I felt like a jerk after all my years of having studied and taught this stuff, but I did get a warm feeling knowing that my son has surpassed me by light years. And the boy did console me with the following words after pointing out my mistake:

<“It's not an easy problem though, don't think it was obvious for me.”>

But was he just being kind to an old guy on the skids? (<)

________________

*Ken – July 7, 2006*