Random Walk Probability Calculator

About this calculator

The random walk probability calculator computes, for a simple 1D random walk of n steps where each step moves +1 with probability p and −1 with probability q = 1 − p, the probability of ending at a specific position k. Since ending at position k after n steps requires exactly (n+k)/2 forward steps and (n−k)/2 backward steps, the answer follows the binomial distribution: P(k) = C(n, (n+k)/2) × p^((n+k)/2) × q^((n−k)/2).

The random walk is one of the most widely reused models in applied math: it describes the diffusion of a particle in Brownian motion, is the discrete-time skeleton behind stock-price models like geometric Brownian motion, and shows up in gambler's-ruin problems in probability theory. Statistics and finance students, and researchers modeling stochastic processes use this calculator to check the closed-form binomial result against a simulation or textbook problem.