Q: The "house" rolls two 20-sided dice and the "player" rolls one 20-sided die. If the player rolls a number on his die between the two numbers the house rolled, then the player wins. Otherwise, the house wins (including ties). What are the probabilities of the player winning?

A: It is easily seen that if any two of the three dice agree that the house wins. The probability that this does not happen is 19*18/(20*20). If the three numbers are different, the probability of winning is 1/3. So the chance of winning is 19*18/(20*20*3) = 3*19/200 = 57/200.

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