Paper 2, Section II, F

Lionel and Cristiana have $a$ and $b$ million pounds, respectively, where $a, b \in \mathbb{N}$. They play a series of independent football games in each of which the winner receives one million pounds from the loser (a draw cannot occur). They stop when one player has lost his or her entire fortune. Lionel wins each game with probability $0<p<1$ and Cristiana wins with probability $q=1-p$, where $p \neq q$. Find the expected number of games before they stop playing.

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