Galton board
Sir Francis Galton’s 1889 device for demonstrating the central limit theorem. Balls drop from a funnel through a triangular array of pegs. At each peg, a ball bounces randomly left or right. The balls accumulate in bins at the bottom, forming a binomial distribution that converges to the bell curve as the number of balls grows. Order from randomness.
P(k) = C(n,k) · pk · (1−p)n−k | n = peg rows, p = 0.5
Galton’s bean machine (1889)
Sir Francis Galton built his “quincunx” to make an abstract theorem tangible. The device is simple: a vertical board with a funnel at the top, a triangular array of evenly spaced pegs below, and a row of bins at the bottom. Balls are dropped through the funnel. At each peg, a ball bounces either left or right with roughly equal probability. After passing through n rows of pegs, each ball lands in one of n + 1 bins. Galton first described the device in his 1889 book Natural Inheritance, using it to illustrate how the aggregate behavior of many independent random events produces a predictable, bell-shaped distribution.
The central limit theorem
The Galton board is a physical proof of the central limit theorem, one of the most powerful results in probability theory. The theorem states that the sum of a large number of independent random variables — regardless of their individual distributions — will tend toward a normal (Gaussian) distribution. Each peg in the board represents one independent random variable: a coin flip that sends the ball left or right. After n such flips, the ball’s horizontal position is the sum of n independent ±1 random variables. The distribution of these sums is the binomial distribution B(n, 0.5), and as n grows, the binomial converges to the normal. This is why the histogram in the bins approaches the bell curve — it is not a coincidence but a mathematical necessity.
Pascal’s triangle
There is a beautiful connection between the Galton board and Pascal’s triangle. The number of distinct paths a ball can take to reach the k-th bin after n rows of pegs is exactly C(n, k) — the binomial coefficient, which is the entry in the n-th row and k-th column of Pascal’s triangle. The pegs physically compute Pascal’s triangle: each peg splits incoming paths into two, exactly mirroring the addition rule C(n, k) = C(n−1, k−1) + C(n−1, k). The histogram you see forming at the bottom is a direct readout of a row of Pascal’s triangle, normalized by 2n.
Real-world applications
The bell curve that emerges from the Galton board appears throughout science and everyday life. Measurement error: every physical measurement is the sum of many small independent sources of noise, so measurement errors are normally distributed. Polling and surveys: the margin of error in election polls comes directly from the central limit theorem applied to sample means. Quality control: manufacturing tolerances are set using the assumption that defect rates follow a normal distribution around the target specification. Financial models: the Black-Scholes option pricing model assumes log-normal returns, a direct descendant of the central limit theorem. The bell curve is not just a mathematical curiosity — it is the default distribution of nature whenever many small independent effects combine.
The surprising universality of the bell curve
What makes the central limit theorem so remarkable is its universality. It does not matter whether the individual random variables are coin flips, dice rolls, exponential waiting times, or any other distribution with finite variance. The sum always converges to the same shape. This universality is why the normal distribution appears in contexts that seem to have nothing in common: the heights of people, the velocities of gas molecules, IQ scores, the positions of stars in a galaxy cluster, and the error in a GPS measurement. Galton himself was struck by this, writing that the law of frequency of error “reigns with serenity and in complete self-effacement amidst the wildest confusion.” The Galton board makes this visible: pure chaos at the level of individual balls, pure order at the level of the collective.
Connection to collective behavior research
The Galton board illustrates a theme central to Joshua Becker’s research on collective intelligence: aggregate-level patterns that no individual contributes to intentionally. No single ball “aims” for the center of the distribution. No ball knows where the other balls have landed. Yet the collective pattern is strikingly regular and predictable. This is the same principle that operates in crowd wisdom, market prices, and social norms — the structure of the aggregate emerges from the independence and multiplicity of the individual decisions, not from any coordination or intention. The Schelling segregation experiment shows the opposite face of this coin: mild individual preferences amplified into extreme collective patterns. Here, individual randomness is averaged into collective regularity. Both are consequences of the same mathematics.