Symmetric Walk Ends at Zero

This quant interview question is about a symmetric random walk with absorbing barriers and the expected number of visits to a distant state before absorption. It sits at the intersection of probability, stochastic processes, and martingale-style thinking, all of which are core to serious quant prep and high-end quant interviews. The setup forces you to reason about paths, not just endpoints.

It trains your understanding of Markov chains, hitting probabilities, geometric-type repetition of events, and the structure of random walks with absorbing states. You must keep track of conditioning on intermediate states, realize when the process effectively restarts, and translate an intuitive path picture into precise expectations.

This matters for quant interviews because pricing, risk, and algorithmic trading models all rely on similar ideas: path-dependent events, barrier-like features, and repeated returns to key levels. Being comfortable with such random walk expectations signals that you can handle more advanced stochastic modeling under pressure in real interviews.

What it tests

In symmetric random walks with absorbing barriers, the expected number of visits to a particular state before absorption can be understood through first-step analysis and the Markov property. The walk's future is independent of its past, given its current position, so the process 'restarts' each time a state is revisited. The probability of reaching one absorbing barrier before another from any starting point is linear in the starting position, leading to simple ratios like $a/b$. The number of times a non-absorbing state is visited before absorption is governed by geometric distributions, because after each visit, the process either gets absorbed or returns, with fixed probabilities. This structure arises from the memoryless property of the walk and the fact that, after each visit, the situation is statistically identical to the previous one, except possibly for the starting position.