Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 1 / b / Solution

Let $\tau =T_a\wedge T_b$. The walk exits the finite interval $[a,b]$ almost surely, and the stopped martingale $\varphi (S_{n\wedge \tau })$ is bounded between ${\lambda }^b$ and ${\lambda }^a$. The optional stopping theorem and bounded convergence theorem giveSolving for the first probability gives the biased gambler's ruin probability