Expected Gambler's Duration I

A gambler starts with </code>k$ , where $k$ is an integer. Each round of a game, the gambler has probability $0 < p < 1$ of winning, independently between rounds, while the casino has probability $1-p$ of winning. Assume $p \neq 1/2$ . The winner of each round receives $$1$ from the loser. If the gambler has either $$0$ or </code>m$ , where $m > k$ , at any point, they leave the table. Assuming that the casino has unlimited money, find the expected number of rounds that the gambler plays. Report the answer to the nearest integer when $p = 0.45, k = 60,$ and $m = 100$ .