Drop a ball at the top. It bounces off each peg, going left or right with equal probability. After n rows, where does it land? Run thousands of balls and the bottom histogram approaches the Gaussian bell curve — the Central Limit Theorem made physical. Bias the probability and the peak shifts.
why a bell curve emerges
Each ball makes n independent left/right choices. The final position is (R − L) / 1 where R + L = n. The number of paths reaching bin k is the binomial coefficient C(n, k). With probability p of going right each time, the distribution is
P(k right) = C(n,k) · p^k · (1−p)^(n−k)
This is the Binomial distribution. As n grows, by the Central Limit Theorem, it converges to the Gaussian
N(μ = np, σ² = np(1−p))
For p = 0.5 (unbiased pegs), the Gaussian is centered. Slide bias to 80%: the peak moves right, the variance shrinks. Any sum of independent random variables — regardless of their individual distributions — becomes Gaussian under repetition. This is the deepest reason the bell curve appears everywhere in nature: measurement errors, particle velocities (Maxwell-Boltzmann), test scores, heights, financial returns. The Galton board is the simplest physical demonstration.