Random Walks in Quant Interviews

A simple random walk steps +1 or −1 each round. It is the discrete skeleton of Brownian motion, the model behind gambler's ruin, and the setting for a large share of stochastic-process interview questions. Four facts plus one trick cover most of what gets asked.

The four facts

• Hitting probabilities. From $a$, the chance of hitting $N$ before 0 is $a/N$ (fair) — see gambler's ruin for the biased version.
• Exit times. Expected time to leave $[0, N]$ from $a$ is $a(N-a)$. A walk on an interval of width $w$ takes on the order of $w^2$ steps to cross it — the square-root scaling that underlies all diffusion intuition.
• Recurrence. The symmetric walk returns to 0 with probability 1 in 1D and 2D, but NOT in 3D ("a drunk man finds his way home; a drunk bird may be lost forever") — a classic conceptual question. But the expected return time is infinite even in 1D — the standard follow-up.
• Position distribution. After $n$ steps, position is a shifted binomial: mean 0, variance $n$, standard deviation $\sqrt{n}$. "Where is the walk after 100 steps, roughly?" — within about ±10 with two-thirds probability.

The trick: the reflection principle

To count paths that touch a barrier, reflect the path after the first touch. It converts "paths from A to B that hit level $m$" into "all paths from A' (the reflection) to B" — giving the ballot problem, the distribution of the maximum, and first-passage counts in one move. If you state "reflect at the first touch" cleanly, most interviewers consider the question answered.

Worked micro-example

Fair walk from 0; probability it reaches +1 within 3 steps? Paths of length 3: the walk reaches +1 immediately with the first up-step ($1/2$); if it steps down first (prob $1/2$), it needs two up-steps to be back at 0... enumerate: sequences starting D that touch +1 by step 3: DUU only reaches 0 at step 3 — never +1. So the answer is the probability of U on step 1 = ... plus DU? D then U reaches 0, then U reaches +1 at step 3: DUU ends at +1! So P = P(U) + P(DUU) = 1/2 + 1/8 = 5/8. Enumerating carefully beats waving at formulas — interviewers watch for that discipline on small cases.

Practice with solutions

• Expected steps to reach position 10
• Biased walk reaching +1
• Absorption on a finite walk
• The stochastic-process bank — Brownian motion and beyond.