Two n-sided dice, with numbers ranging from 0 to n-1, are rolled. The first die represents the first digit of a base-n number, and the second represents the second digit of said base-n number. (In other words, the two dice rolls, if they are a and b, give the number a+b*n). Assuming that both dice are rolled concurrently, and each die is rolled until it rolls an n-1, at which point the other die continues to be rolled until it also rolls an n-1, what is the average number of rolls necessary until both dice reach their maximum value?
I... don't really know.
The Attempt at a Solution
I've never taken a probability class, so I have no idea what to do here. The idea came to me while I was playing Dungeons and Dragons, and the original idea involved 10-sided dice, but I thought the general result might be more fun to see. Could someone help me with this problem, tell me what I need to do to solve it?