Probabilistic Structure of the Galton Board

The Galton board (or quincunx) is a mechanical device that provides a physical illustration of fundamental ideas in probability theory. A ball falls through a triangular array of pegs, and at each peg it is deflected either to the left or to the right. Under the natural assumption that each deflection is independent and occurs with fixed probability, the motion of a single ball can be modelled as a sequence of independent Bernoulli trials.
The final horizontal position (bin index) of the ball is therefore the sum of these Bernoulli random variables and follows a binomial distribution. When the number of rows of pegs is moderately large, the binomial distribution is well approximated by a normal distribution, explaining the characteristic bell-shaped curve observed in the device.
When many balls are dropped independently, the empirical histogram of landing positions converges to the underlying binomial probabilities by the Law of Large Numbers. Thus, the Galton board illustrates several central concepts of probability theory: Bernoulli trials, the binomial distribution, the normal approximation, and convergence of empirical frequencies.

Model of a Single Ball

Normal Approximation (De Moivre–Laplace)

Many Balls: Empirical Distribution

CLT Over Balls

Conceptual Summary