The Galton Board

What just happened?

The Galton board (Galton board, quincunx) is a device invented by Francis Galton in 1889. A ball falling through a grid of 8 rows of pegs bounces randomly left or right at each peg with probability $\tfrac{1}{2}$. The number of right moves determines which bin it lands in.

The binomial distribution

The number of right moves follows the binomial distribution: the probability of landing in bin k is

The numerator $\binom{n}{k}$ is the number of distinct paths leading to bin k (the binomial coefficient, a row of Pascal’s triangle). The most paths lead to the center bins, so they are the most likely — hence the bars are tallest there.

Order emerges from chaos

Although each ball’s path is purely random, after summing thousands of balls the bars settle into a regular bell curve (Gaussian). This happens because the binomial distribution for large n tends to the normal distribution — this is the central limit theorem:

with center $\mu = \tfrac{n}{2}$ and standard deviation $\sigma = \tfrac{\sqrt{n}}{2}$. In the panel on the right you can see how the exact binomial probabilities match the normal-density markers — the better, the larger n is.

Why it matters

The Galton board is a beautiful, tangible illustration of how precise statistical order emerges from the pure chaos of individual events. It is a classic model showing the ubiquity of the normal distribution in nature: human heights, measurement errors, test scores — everywhere the outcome is the sum of many small, independent influences.

Why this matters for your exam:

The binomial distribution $P(k)=\binom{n}{k}p^k(1-p)^{n-k}$ (here $p=\tfrac12$) describes the number of successes in n independent trials — the basis of Bernoulli-scheme problems. When there are many trials, we approximate it with the normal distribution. The trick: when a problem has “n independent trials”, count via $\binom{n}{k}$, and for large n expect a “bell”.

Remember: the probability of bin k is $C(n,k)/2^n$ (the binomial distribution), and the sum of many random bounces settles into a bell curve — the normal distribution (the central limit theorem).