Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 2 / 10E / b / i / Solution

Let $h(x)$ be the probability that the simple symmetric random walk starting at $x\in \{0,\dots ,n\}$ hits zero before $n$. The first-step recurrence isThe recurrence says that $h$ is affine, and the boundary values determine it: