GALTON BOARD

Interactive Galton Board Simulation

Watch 1,400 balls fall through a physics-based Galton board, also called a bean machine or quincunx. Each collision sends a ball along a different path; together, the final bins reveal the order hidden inside many random events.

How the Galton board works

At each peg, an idealized ball has an equal probability of moving left or right. With n rows, every ball makes n binary choices, and its final bin depends on the number of rightward moves.

p=0.5,1−p=0.5p = 0.5, \quad 1-p = 0.5

Why the bins follow a binomial distribution

Every left-or-right collision is a Bernoulli trial. The number of rightward moves therefore follows a binomial distribution: n is the number of rows, k is the number of right turns, and p is the probability of moving right. The probability of landing in bin k is:

P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k} p^k (1-p)^{n-k}

How the bell curve emerges

As the number of rows grows, the binomial shape is increasingly well approximated by the normal distribution. The Galton board visualizes the normal approximation to the binomial and gives an intuitive illustration of the central limit theorem: repeated independent contributions aggregate into an approximately normal shape.

f(x)=1σ2πe−12(x−μσ)2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

For the idealized board, the binomial mean is μ = np and the variance is σ² = np(1−p). Individual paths remain unpredictable, while the aggregate pattern becomes remarkably stable.

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