Monday, May 28, 2007

Monty Hall Revisited

My middle school son came to me with this twist on the famous Monty Hall/Let's Make a Deal probability problem. I am looking for the answer to the problem, and want the clearest most succinct explanation.

The original Monty Hall problem goes like this: There are three doors, and behind one is a prize. A contestant picks a door, but does not get to look behind it yet. Monty Hall, the master of ceremony (who knows which door holds the prize), opens one door, showing that there is nothing behind it. He then gives the contestant the chance to switch doors: The question is, should one switch?

(There are game theoretic aspects to this problem that are often ignored. Let's assume that Monty Hall ALWAYS opens an empty door and gives the contestant the chance to switch, no matter if the contestant currently has the right door or not.)

The answer is that one should switch doors. Not the most intuitive probability exercise for many people, but correct, given our parenthetic note above.

Now here is the twist. An exam in school will be either Monday , Tuesday or Wednesday, and the teacher has not said which. A student is assessing the odds of what day the test will be in order to study the night before. The day picked by the student is Wednesday, and she has arranged her schedule to study Tuesday night. Monday comes, and the teacher announces that the test will not be that day. So...should our student switch her pick, just like the contestant in Let's Make a Deal?

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