The "house" rolls two 20-sided dice and the "player" rolls one 20-sided die.

probability/icos

Problem

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?

Show solution

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.