Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 11F / a

Let $h_k$ be the hitting probability of the right endpoint before the left endpoint, starting at $k$. A surviving vase corresponds to that right-endpoint event. Conditioning on the first step of the simple symmetric random walk givesThe Markov property makes the remaining probability depend only on the position after that step. These absorbing endpoint values distinguish the event of interest from its complement.