The Physical Bell Curve Machine
The Galton board — also known as the bean machine or quincunx — is one of the most elegant demonstrations in all of mathematics. Balls are dropped from a single point at the top and cascade through rows of evenly-spaced pegs. At each peg, a ball bounces randomly left or right with equal probability. Despite this randomness, the balls accumulate at the bottom in a remarkably predictable pattern: the bell curve.
Why Randomness Creates Order
Each ball's final position is the sum of many small random displacements — one at each row of pegs. The Central Limit Theorem guarantees that such sums converge to a normal distribution, regardless of the underlying probability. This is why the bell curve appears everywhere in nature: heights, test scores, measurement errors, and thermal fluctuations all arise from the accumulation of many small independent effects.
Reading the Simulation
The pegs are arranged in a triangular array. Each ball falls from the top and bounces left or right at each peg, accumulating in bins at the bottom. The histogram of bin heights forms the characteristic bell shape. The cyan overlay shows the theoretical normal curve for comparison. Adjust the bias parameter to see how shifting the left/right probability moves the peak.
From Binomial to Normal
With n rows and right-probability p, each ball follows a binomial distribution B(n, p). For the default settings (n=12, p=0.5), this is symmetric with mean 6 and standard deviation √3 ≈ 1.73. As the number of rows increases, the binomial distribution becomes indistinguishable from a normal distribution — this convergence is one of the most important results in probability theory.