Random walk

A particle starts at position 0 and at each integer time step moves one unit right (with probability p) or one unit left (with probability 1 − p). Where does it end up after t steps?

±√t envelope (1σ)

steps (T) 200 walkers 20 p (step right) 0.50

20 independent walkers, each step ±1. Dashed envelope: ±

t

(standard deviation grows as

t

, not t). Bias: drift = 0.00 per step.

The √t law

For an unbiased walk (p = 0.5), the expected position is always 0 — steps cancel on average. But the spread grows:

Var (X_{t}) = t

Std (X_{t}) = t

The dashed envelope in the chart shows ±√t. After 100 steps, walkers are typically within ±10 of the origin. After 10,000 steps, within ±100. Spread grows as the square root of time, not linearly.

Bias

With p ≠ 0.5, each step has a net drift of 2p − 1 per step:

E [X_{t}] = (2 p - 1) t

The amber line shows this drift. A small bias (say p = 0.52) barely moves the mean drift line — but over thousands of steps, it dominates. This is the mathematics behind the casino edge and why even a 2% house advantage reliably extracts money over time.

Diffusion and Brownian motion

The random walk is the discrete ancestor of Brownian motion. As the step size shrinks and steps become more frequent, the walk converges to a continuous Gaussian process where position after time t is distributed as Normal(0, t). The same √t scaling survives in the limit — it is a fundamental feature of diffusion, not an artifact of the discrete model.

The Central Limit Theorem explains why: the position after t steps is a sum of t independent ±1 variables, and sums of independent variables tend toward Normal distributions as t grows.