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Programs and Probabilities


In response to Brian Hayes’ postscript “Monty Hall Redux” (Sept-Oct 2008):

I had the wrong Monty Hall answer at first until I read about the computer simulation and decided to rethink the problem, which then prompted me to develop a new understanding. When confronted with an anomaly like conflicting results from a computer model, we all respond differently (locating the error in self vs. other/other's model) because even people who share a belief in logic as a means of solving problems have widely varying degrees of faith in their own powers of reason. Dissent is healthy but digging in one’s heels in the face of compelling evidence hinders progress, and the line between the two isn’t always clear. (Forget Monty Hall, how do you persuade others that humility is a virtue?)

Rather than continuing a fruitless debate or acknowledging dissent and moving on without resolution, a third option is to try translating the problem into different terms. In this case we can use the language of human behavior, recognizing that Monty Hall is actively responding to the player’s initial choice. To make it even clearer we can imagine that Monty Hall wants the player to lose; he’s greedy and wants to keep the prize.

Consider the only two possibilities: the player’s initial door selection either has or does not have the prize behind it. As a player, you don’t know if you’ve selected the door with the prize. But Monty Hall does “know” and acts accordingly. (Even if this knowledge isn’t explicitly stated, it’s implicit in Monty Hall’s behavior of opening only unchosen doors, coupled with his knowledge of where the prize is.)

Case 1 - If your chosen door doesn’t have the prize behind it (probability of 2/3), then Monty Hall has no choice. He must open the lone door that was neither selected by you nor has the prize, and hope that you don’t switch. Because in this case, switching will always give you the prize.

Case 2 - If your chosen door does have the prize (probability of 1/3), then Monty Hall has a choice; he may open either of the doors you didn’t select. But of course it doesn’t matter which door he opens. He’s just hoping you’ll switch, because if you switch you’ll never get the prize.

Since you don’t know whether the door you chose has the prize behind it (determining whether you’re operating in Case 1 or Case 2), you should switch since it’s twice as likely that you’re in Case 1. Twice as likely that Monty Hall had no choice when he opened the door he did. The “always” and “never” consequences to switching make it mathematically simple - switching gives you the prize 2/3 of the time.

posted by Barbara Blossom
August 9, 2008


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