Drop walkers through a Galton board: each step goes up with probability p and down otherwise. The pile-up follows the binomial law and, for many steps, a normal curve.
The random walk is the discrete starting point of stochastic models: add up independent ±1 coin-flip steps and watch the sum wander. This lab simulates seeded walks, rescales them by √n towards Brownian motion, measures how the spread grows with time and shows the central limit theorem turning sums of any steps into a normal distribution.
A sum of independent steps that are +1 or −1 with equal probability. Its expected value stays 0, but its variance grows by 1 every step, so after n steps the typical distance from the start is √n.
Independent steps partly cancel. Variances of independent steps add, so variance grows linearly in time and the standard deviation, its square root, grows like √t.
Shrink the step size to 1/√n while taking n steps per unit of time. As n grows, the rescaled walk converges to Brownian motion (Donsker's principle); the Brownian Motion lab picks up from there.
The standardised sum of many independent steps with finite variance is approximately normal, whatever the distribution of a single step. That is why the endpoint of a long walk looks Gaussian.