The Galton Board: Binomial Chance Becomes a Bell Curve

A machine that draws the binomial distribution

The Galton board — also called a bean machine, after Francis Galton, who built one in the 1890s to make probability visible — is a vertical board studded with rows of pegs arranged in a triangle. A ball is dropped at the top, hits the first peg, and bounces left or right with roughly equal probability. It then falls onto the next row and repeats the choice, row after row, until it lands in one of the bins along the bottom.

Each bounce is an independent coin flip. After n rows, a ball's final bin position is the number of right-bounces it made out of n tries — exactly the definition of a binomial random variable with parameters n and p = 0.5. Drop a few thousand balls and the bins fill up in proportion to the binomial probabilities, and the classic bell-shaped pile of balls is not a metaphor for the distribution — it is the distribution, built physically one ball at a time.

Pascal's triangle is the exact answer

The probability that a ball ends in bin k after n rows is given exactly by the binomial formula, and the coefficients are the rows of Pascal's triangle: each entry is the number of distinct left/right paths that lead to that bin.

P(bin = k) = C(n,k) · p^k · (1-p)^(n-k)

n=4 row of Pascal's triangle:  1  4  6  4  1
→ probabilities (p=0.5):    1/16 4/16 6/16 4/16 1/16

With p = 0.5 the triangle is symmetric and the middle bins are the most likely landing spots, simply because there are more distinct paths (more ways to interleave lefts and rights) that reach the centre than the edges. Bin 0 (all lefts) and bin n (all rights) each have exactly one path, so they are always the rarest outcomes.

Why it looks like a normal curve

As n grows, the jagged binomial histogram smooths into the familiar bell curve. This is the central limit theorem in its most literal form: a sum of many small, independent, identically distributed steps (here, ±1 per row) converges to a normal distribution regardless of the details of each step. The de Moivre–Laplace theorem, proved in the 1730s — decades before Gauss — is the special case that shows the binomial distribution itself is well approximated by a normal curve with mean np and standard deviation √(np(1-p)) once n is reasonably large.

That variance formula also explains the board's characteristic shape: the spread of the pile grows only as √n, not as n, so adding more rows makes the bell noticeably narrower relative to its width — a visual demonstration of why averages of more measurements are more reliable.

From toy to statistical tool

The bean machine was not built as a toy. Galton used it to argue that traits shaped by many small independent hereditary and environmental influences — height, for instance — should be normally distributed in a population, an idea central to the birth of biometrics and regression analysis. The same peg-board logic underlies random walks, diffusion of particles (Brownian motion is its continuous-time limit), and quality-control charts that expect measurement error to cluster in a bell around a true value.

Skew the board by biasing p away from 0.5 — a peg alignment that favours one side — and the distribution stops being symmetric but the same binomial machinery still applies; only the mean np shifts and the normal approximation becomes asymmetric until n is large enough to smooth it out again.

Frequently asked questions

Try it live

Everything above runs in your browser — open Galton Board and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.