I read about this game as a kid, but my maths was never up to solving it:
The score starts at zero. Take a shuffled pack of cards and keep dealing face up until you reach the first Ace, at which the score becomes 1. Deal on until you reach the next 2, at which the score becomes 2, although you may not reach this if all the 2s came before the first Ace. If you reach 2, deal on until you reach the first 3, at which, if you reach it, the score becomes 3, and so on. What is the most likely final score? And how do you calculate the probability of any particular score?
I once wrote a program that performed this routine millions of times on randomised packs with different numbers of suits up to about 10. To my surprise, the most likely score for any pack seemed empirically to always be the same as the number of suits in the pack. I would love to see this proved, though it is beyond my powers.