This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
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
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.