Galton Board

6 min read#probability

A board of pegs that turns a stream of falling balls into a bell-shaped pile — the central limit theorem made physical.

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Galton Board

A Galton board — also called a bean machine or quincunx — is the central limit theorem you can hold in your hands. Balls drop from a single point at the top and tumble through a triangular lattice of pegs. At each peg a ball bounces left or right with equal chance, then falls to the next row and bounces again. After many rows the balls collect in bins along the bottom — and they always pile up into the same shape: a bell curve.

Invented by Sir Francis Galton in the 1870s, it is a mechanical proof that order emerges from accumulated randomness. No ball's path is predictable, yet the pile is utterly reliable.

Each ball is a random walk

Follow one ball. At each of the n rows it steps left (-1) or right (+1) with probability \tfrac{1}{2} — exactly a Random Walk. Its final bin is the sum of those n independent \pm 1 choices, so the number of balls landing in each bin follows a binomial distribution. And a binomial with many trials is, by the Central Limit Theorem, approximately normal:

\text{bin position} = \sum_{i=1}^{n} s_i, \qquad s_i = \pm 1 \text{ each with prob } \tfrac{1}{2}.
(1)

The center bins fill fastest because there are many left/right combinations that cancel out to land there, but only one path — all-right — that reaches the far edge. Counting paths is counting the binomial coefficients of Pascal's triangle, and their smooth envelope is the bell.

Watch the bell build

Below, balls fall one after another, bouncing left or right at each peg and settling into bins at the bottom. Any single ball's route is a coin-flip cascade — but watch the bins. They fill into the unmistakable bell curve, taller in the middle, thinner at the edges, growing smoother with every ball.

A Galton board: balls bounce left/right at each peg (a random walk) and stack into bins. The pile grows into a binomial/normal bell curve — the central limit theorem made physical. Resets when the bins fill.

A machine you can read two ways

The Galton board is a rare object that is simultaneously a physical experiment and a mathematical proof. As physics, it is gravity and elastic bounces. As mathematics, it is the binomial distribution converging to the normal — the Central Limit Theorem and the Probability Distribution of a Random Walk stacked into a literal pile. Galton built it to argue that complex, bell-shaped variation in nature need not have a complex cause: many tiny independent nudges suffice.

See also

Related topics

  • Probability DistributionA complete description of how likelihood is spread across the possible outcomes of a random quantity.
  • Central Limit TheoremThe sum or average of many independent random quantities tends to a normal distribution, no matter what shape the individual quantities have.
  • Random WalkA path built by adding up independent random steps; its typical distance from the start grows like the square root of the number of steps.
  • Law of Large NumbersAs you collect more independent samples, their running average converges to the true expected value.
  • Probability & Random ProcessesHow randomness, far from being formless, organizes itself into sharp and predictable structure — bell curves, steady states, and laws of large numbers.
  • Brownian MotionThe continuous-time limit of a random walk — the erratic, jittery path traced by a particle buffeted by countless tiny random kicks.

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