experiment 035 · probability · statistics

Galton Board

Drop a ball into a triangular array of pegs. At each peg, the ball bounces left or right with equal probability. What seems like chaotic tumbling at the micro scale transforms into elegant order at the macro scale. As thousands of balls accumulate in the bins below, the binomial distribution emerges—a live demonstration of the Central Limit Theorem.

The system

The Galton Board (or quincunx) was invented by Sir Francis Galton to demonstrate the Central Limit Theorem. Each ball undergoes a series of independent Bernoulli trials. For a board with $n$ rows of pegs, the number of times a ball bounces right follows a binomial distribution:

$$ P(k) = \binom{n}{k} p^k (1-p)^{n-k} $$

With symmetric pegs, the probability of bouncing right is $p = 0.5$. As $n$ becomes large, the binomial distribution approaches the normal (Gaussian) distribution:

$$ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} $$

Numerical note

This is a continuous 2D rigid-body simulation, not a discrete state machine. The balls have real mass, radius, position, and velocity, integrated via the semi-implicit Euler method. Collisions with pegs are resolved using impulse-based physics, introducing natural microscopic variance that drives the macroscopic statistics.

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