Problem 1:

At the beginning of a game an urn contains 100 white marbles and 100 black marbles. Each turn consists of randomly taking out two marbles. If they are both the same color, a white one is put back. If they are different colors, a black one is put back. Thus after each turn there is one less marble then before.

If you had to bet on which color the last marble would be, which color would you choose and what would be the probability of winning the bet?

Solution: The key to this problem is to realize that it is not really a probability problem at all. Note that no matter what combination of marbles is taken out, the parity of the black marbles never changes. Since we start with an even number of black marbles, there will always be an even number of black marbles (zero is an even number too). The number of marbles goes down by 1 each turn, we will get down to 1 marble after 199 turns. Since 1 is an odd number, this last marble must be white. Go ahead and bet all the money you have!

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