The Binomial Distribution and the Galton Board

Bernoulli trials: the building block

A Bernoulli trial is any random experiment with exactly two possible outcomes, conventionally labelled "success" and "failure", where success occurs with fixed probability p and failure with probability 1 − p. A single coin flip, one shot on goal, whether a single component fails within a year, whether a single visitor to a website makes a purchase — all of these are Bernoulli trials once you fix what counts as "success".

The binomial distribution describes the total number of successes, X, when you repeat a Bernoulli trial n times independently, with the same success probability p on every trial. Two conditions are doing real work in that sentence: independence (one trial's outcome cannot influence another's) and a constant p (the success probability cannot drift between trials). Both assumptions are what let you multiply probabilities together cleanly; violate either one — say, by sampling without replacement from a small finite population — and the true distribution becomes the related but distinct hypergeometric distribution instead.

The binomial probability mass function

The probability of observing exactly k successes in n trials is given by

P(X = k) = C(n, k) · p^k · (1 − p)^n−k

where C(n, k) = n! / (k!(n−k)!) is the binomial coefficient, counting the number of distinct orderings of successes and failures that produce exactly k successes among n trials. The term p^k(1−p)^n−k is the probability of any one specific sequence with k successes; multiplying by C(n, k) accounts for the fact that those k successes could have landed on any of the C(n, k) possible positions among the n trials. This is exactly the same combinatorial logic that appears in Pascal's triangle, where each row lists the binomial coefficients for a given n.

The mean of a binomial distribution is E[X] = np, and its variance is Var(X) = np(1 − p), so its standard deviation is √(np(1−p)). Both formulas make intuitive sense at the extremes: when p is near 0 or 1, outcomes are nearly deterministic and the variance shrinks toward zero; when p = 0.5, the variance is maximised for a given n, reflecting maximum uncertainty about each individual trial.

The Galton board: a physical proof

The Galton board (also called a bean machine or quincunx), invented by Francis Galton in the 1890s, is a vertical board studded with rows of pegs arranged in a triangular grid, with collection bins along the bottom. A ball dropped from the top strikes each peg and bounces left or right with roughly equal probability, then falls to the next row and repeats the process. After passing through n rows of pegs, the ball's final bin position is determined entirely by how many times it bounced right versus left — which is precisely a sum of n independent Bernoulli trials with p = 0.5.

Drop hundreds of balls through the same board and the height of beads accumulated in each bin traces out, almost exactly, the binomial probability mass function for n trials and p = 0.5. This is why the Galton board is such a compelling teaching tool: it is not a simulation or an approximation drawn on paper, it is the binomial distribution being generated mechanically, bead by bead, in front of your eyes. It also foreshadows the Central Limit Theorem, because as the number of peg rows n grows large, the increasingly smooth, bell-shaped outline of the bead pile converges toward a normal distribution — the binomial distribution's continuous limiting shape.

How n and p shape the distribution

When p = 0.5, the binomial distribution is perfectly symmetric around its mean np, regardless of n, because success and failure play interchangeable roles. As p moves away from 0.5 toward either extreme, the distribution becomes skewed: right-skewed for small p (most trials fail, with an occasional cluster of successes), left-skewed for large p. Increasing n while holding p fixed makes the distribution progressively more symmetric and bell-shaped even for skewed p, because of the same central-limit effect visible in the Galton board — a useful rule of thumb is that the normal approximation to the binomial becomes reasonable once both np ≥ 5 and n(1−p) ≥ 5.

When to reach for the binomial distribution

The binomial model applies whenever you are counting successes across a fixed, known number of independent identical trials: the number of heads in 20 coin flips, the number of defective items in a batch of 100 sampled with replacement, the number of patients who respond to a treatment out of a fixed trial cohort (assuming a constant response probability across patients), or the number of free throws made out of a fixed number of attempts. It breaks down when the number of trials is itself random, when the trials are not independent (for example, a shooter whose confidence and accuracy change after a make or a miss), or when the success probability genuinely varies from trial to trial — in that last case, a Beta-Binomial or other mixture model is usually more appropriate. Recognising which of these conditions actually holds in a given dataset is often more important than getting the arithmetic of C(n,k) right.