1. Playing a game at a fair, you are asked to choose one of four boxes, which are identical in appearance.
One of the boxes contains a prize and the other three are empty. Once you select a box, the carnie running the game opens one of the other boxes, shows you that it is empty, and gives you the chance to change your mind, i.e. to pick a box other than the one you selected first. Will changing your selection improve your chances of winning or not? Give a complete explanation!
CLUE? No one expects a snakeless mathematical inquisition to come to this hall!

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Answer: Switching your selection improves your chance of winning. We give two answers.

Answer 1: Suppose that you have chosen box 1. With probability...
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