Tourney Ending II

Alice and Bob play in a tennis tournament such that the winner of each round receives $1$ point, while the loser loses $1$ point. The tournament ends as soon as one of the players reaches $2$ points or $2n$ games have been played, whichever comes first. Alice has probability $0 < p < 1$ of winning each round. Let $E(n,p)$ be the expected number of games played in the tournament. If it exists, compute $\displaystyle \lim_{n \rightarrow \infty} E(n,p)$ when $p = 1/3$.