Expected number of shots for a game to end

Question

A basketball player plays a shooting game. He gets +1 point if he scores a basket and -2 points if he misses. He starts with 0 points. The game ends when the player reaches +10 or -10. What is the expected number of shots taken for a game to end, given a player scores a basket with probability p.

Answer 1

For $-11\le n\le 10$, let $e_n$ denote the expected number of rounds until a score $\le -10$ or $\ge 10$ is reached when starting with a score of $n$. Clearly, $e_{-11}=e_{-10}=e_{10}=0$ while for $-10<n<10$ we have $e_n=1+pe_{n+1}+(1-p)e_{n-2}$. This gives you $19$ linear equations in $19$ unknowns.