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Tuesday, July 29, 2008

Bizarre Boxes

Someone shows you two boxes and he tells you that one of these boxes contains two times as much as the other one, but he does not tell you which one this is. He lets you choose one of these boxes, and opens it. It turns out to be filled with $10. Now he gives you the opportunity to choose the other box instead of the current one (and skip the $10 of the first box), because the second box could contain twice as much (i.e. $20).

The Question: Should you choose the second box, or should you stick to your first choice to maximize the expected amount of money?

A Hint : If you have $10, and you could double this with a chance of 1/2, or half it with a chance of 1/2, one would expect an average of 1/2 * $20 + 1/2 * $5 = $12.5 (so you would expect to gain $2.5)!...


Answer: The Solution: No, there is no reason to change your choice.

An explanation: The hint that is given is misleading! In this puzzle, it is not straightforward to calculate the expectation as suggested in the hint. So, just use your common sense, and note that the chance is and stays 50% that your first choice will be the best one. There is no reason at all to change that choice at the moment you find out that the box contains $10...

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