Wednesday, March 14, 2012

The Monty Hall Problem

The Monty Hall Problem is a 'veridical paradox' (something which seems to defy intuition - as with the Achilles and the tortoise paradox). It was invented (discovered?) by Marylin Vos Savant and first published in Parade Magazine in 1990. It is named after US game show host Monty Hall, although other than supplying his name he has nothing to do with the problem.

In the game show you must chose one of three doors. Behind one is the star prize, and behind the other two are (for some reason) goats. You've chosen one of the doors, hoping that the car is behind it, but if you've chosen badly, you get the goat.

Monty then teases you by opening one of the two unselected doors and revealing a goat (he always chooses a losing door), and he offers you the option of switching doors.

The question is: is there an advantage in switching doors? Should you change your choice in order to increase your chances of winning the car.

The intuitive answer is that there is no advantage in changing your mind, since it seems there is now a 50/50 chance that you will win the car. However, it turns out that given this type of choice you should change your mind. I have found a nice program here that proves the point. It is a simulator in which you can run the scenario automatically up to 1000 times, electing either to keep your original choice or change your mind each time.

If this still seems counter-intuitive, this guy explains it much better than I can, in terms of probabilistic outcomes:

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