Abe and Bob are playing a game. In each round of the game, Abe has a $p$ chance of winning, and Bob has a $q = 1 - p$ chance of winning. Abe wins the entire game if he wins at least 2 more rounds than Bob. The game keeps going on forever if no one wins.
What is the chance that Abe will win? Furthermore, how many rounds ($X$) is a game expected to last?
Originally, I thought this was a geometric distribution with $X \sim Geometric(p^2)$, but then I realized that Bob could win too.