Consider a random walk on $\mathbb{Z}/n\mathbb{Z}$ starting at 0. At each step either add 1 or subtract 1, with probability 1/2 each. Let $0\neq i\in\mathbb{Z}/n\mathbb{Z}$. What is the probability $P_n(i)$ that $i$ is the last point to remain unvisited?
Solution. Consider the first time the walk reaches a point adjacent to $i$ (either $i-1$ or $i+1$). The probability that $i$ is the last point to remain unvisited is then the probability that the walk visits the other point adjacent to $i$ before visiting $i$. This probability is independent of $i$. Hence $P_n(i)=1/(n-1)$. (I don't know the origin of this classic problem.)
