First to 10 or bust in symmetric walk

This classic quant interview question is about a symmetric random walk on a finite set of states with absorbing barriers at both ends. The candidate must reason about hitting one extreme before the other, using only the current position as relevant information. It appears in quant prep books and interviews because it connects discrete probability, Markov chains, and boundary behavior in a clean, testable way.

It trains conditional probability intuition in a dynamic setting, comfort with martingale-style reasoning, and the ability to translate a verbal stochastic process into a precise probabilistic object. It also develops intuition for how boundary conditions shape probabilities even when there is no drift.

This matters for quant interviews because many derivatives, risk and algorithmic trading problems reduce to questions about hitting times, absorption, and random walks, both in discrete and continuous time.

What it tests

In a symmetric random walk with absorbing boundaries, the probability of reaching one boundary before the other depends linearly on the starting position relative to the two boundaries. This is because, at each step, the process has no drift, so the expected position remains unchanged except for the effect of absorption at the endpoints. The key is that the expected value of the process is preserved until absorption, and the probability of absorption at a particular boundary is proportional to the distance from the starting point to the opposite boundary. This linearity arises from the fact that the difference equation governing the absorption probabilities is second order and homogeneous, leading to a linear solution in the symmetric case. The underlying structure is that the walk is memoryless and the only information that matters is the current position, not the path taken.