Random Walk on a Number Line

Problem Statement#

The Question#

Interview question: A particle starts at integer position $k$ on the number line, where $0 < k < N$. At each step, it moves right to $k+1$ with probability $p$, or left to $k-1$ with probability $q = 1 - p$. The walk stops when the particle hits either $0$ or $N$. What is the probability it reaches $N$ before $0$? As a follow-up: what is the expected number of steps until th...