Sunday, February 22, 2015

A Tale of Gender and the Monty Haul Problem


This article tells the story of the time Marilyn vos Savant presented the solution to the Monty Haul problem in her column in Parade Magazine, and was widely derided by readers, including some accused her of erring on account of her using "female logic" and other gender-based criticisms.

For those not familiar with the Monty Haul problem, this diagram from Wikipedia provides a helpful summary of the problem and its solution:



Imagine you're a contestant in a game show. The host, Monty Haul, presents you with three closed doors, and asks you to choose one of them. Behind one of the doors is a new car, but behind the other two doors are goats. You choose door #1. Monty Haul then opens a door which he knows has a goat behind it--door #3. He then gives you a choice: do you want to stay with door #1, or switch to door #2?

Many people think that the chance that the car is behind door #1 is the same as the chance that it is behind door #2--namely, a 1 in 2 chance. But, in fact, there is a 1 in 3 chance that the car is behind door #1, and a 2 in 3 chance that the car is behind door #2, so the solution to problem is to switch to door #2.

The diagram above shows why. When you make your initial choice, there is a 1 in 3 chance that the car is behind door #1, and a 2 in 3 chance that the car is behind doors #2 or #3. When Monty Haul opens door #3, which he knows has a got behind it, the chance that the car is behind door #3 drops to 0. This means that there is now a 2 in 3 chance that the car is behind door #3.

It's true that people find the Monty Haul problem to be counter-intuitive in general, but the article presents evidence that Savant was attacked because of her gender in particular. Some of her scathing, verbally abusive critics were academics with PhDs (who later had to eat a nice hefty slice of humble pie).

The comments section of this article is also intriguing; a fellow going by the nomme de web of "RaguxCixot" is valiantly attempting to prove that the standard answer to the Monty Haul problem is false, despite all attempts by well-informed fellow commenters to prove otherwise.

As far as I can tell, RaguxCixot is making a very basic error: he is assuming that there is a chance that Monty Haul will open a door which reveals the car, rather than a door which reveals a goat. In other words, Ragux is ignoring the stated assumptions in the thought experiment in order to reach his conclusion. To be fair to Ragux, when the Monty Haul problem is stated, the assumption that Monty Haul will never open a door to reveal the car is not always clearly or explicitly stated. Nevertheless, Ragux is being so enthusiastic in his stubbornness that he is either exasperatingly dense (given his evident understanding of probability) or else one of the most clever trolls I have ever seen. (There is at least a 1/3 chance of the latter, of course.)
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