Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 9 / b / i / Solution

The expected value of an indicator random variable is the probability of its event. For a simple symmetric random walk absorbed at $0,k$, the gambler's ruin success probability from $i$ is $i/k$. HenceThe boundary values are $0$ at the left exit and $1$ at chamber $k$.