Beating another player's die roll
Imagine this game:
Player A rolls a number of dice, each dice has a set number of sides. For example, there might be a four-sided die with sides 1,2,3, and 4. He sums up all the numbers on his rolled dice and this number is his "power".
Player B then rolls another number of dice with the same rules and gets his "power".
Given the number of dice and the number of their individual sides for each player, how can I calculate the probability of player A's power beating player B's power?
For example, the function A_beats_B may receive as input that player A rolls two 20-sided dice while player B rolls one 20-sided die and one 10-sided die.
Finding the average power of each player is easy, but I've been, so far, unable to discover an algorithm to compute the probability of A beating B.
A complete solution or even a clue that would set me in the right direction is appreciated.