October 7, 2008
Brain Teaser, #6.
One segment of the game show Let's Make a Deal had three doors, behind one of which was a car and behind each of the other two was a goat. The contestant would win whatever was behind the door he/she chose. The contestant chose one of the three doors, but before it was opened, the host opened up a different door that had a goat behind it. To have the greatest chance of winning the car, should the contestant open the door that was his/her original choice, or open the remaining door?
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Hmm. This one's sort of a toughie, but I'm pretty sure you should always switch doors. Note that this doesn't work on the game show, because I'm pretty sure Monty Hall only sometimes opened one of the remaining doors, not always. But in the simplified example you're describing:
You pick a door. You have a 1/3 chance of being right, and a 2/3 chance that the prize is behind one of the doors you didn't pick--so odds are you've got the wrong door. Monty then eliminates a door from the two you didn't pick... but always the WRONG door of the two (or one of the wrong ones, anyway). Meaning that all the choice has been removed from the 2/3 chance. Now your original door now has its 1/3 chance of being right, but the other door has a 2/3 chance of being right. So you should always switch.
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