# What is the probability that Abe will win?

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.

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A couple of closely related questions were asked fairly recently. This link will lead you to the expectation question, which has a link to the probability question. There are many approaches to an answer other than the ones taken. –  André Nicolas Oct 16 '12 at 2:38
"Abe wins the entire game if he wins at least 2 more rounds than Bob" And Bob wins if... what? –  leonbloy Oct 16 '12 at 2:53
Thank you Andre! Also, Bob also wins if he wins at least 2 more rounds than Abe, sorry. –  John Hoffman Oct 16 '12 at 3:10