ODD ODDS
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Your mistake was not realizing that opening Door #3 tells you more about Door #2 than about the door you originally picked. The reason for this is subtle. The host, in picking Door #3, does not choose from the full set of doors but rather from the subset of doors you did not pick. Each subset’s probability of winning does not change but the probability for a particular door in the second subset does. If you don’t get it find a friend who looks like Monte Hall and play 20 rounds. It will soon become obvious which strategy wins most often. –Robert E. Johanson, Chicago
First, though, I feel obliged to eat a bit of crow. The “common sense” answer, the one I gave, is that if you’ve got two doors and one prize, the chances of picking the right door are 50-50. Given certain key assumptions, which we’ll discuss below, this is wrong.
This strategy changes the odds dramatically. In fact, it can be shown that if, two times out of three, Monte opens a door when the contestant has guessed right the first time–a very rational approach on his part–over the long haul the odds of the prize being behind Door #1 versus Door #2 are 50-50.
