Galton Board: Why Random Bounces Make a Bell Curve

The explanation

Which slot a ball lands in depends only on how many times it turned right, and the number of paths with k rights is a binomial coefficient: many in the middle, few at the edges. Add up many small independent nudges and the result is always close to the normal curve. That is the central limit theorem.

01Counting paths: Pascal’s triangle

A path to a peg comes either from the upper left or the upper right, so its count is the sum of the two above: exactly the rule of Pascal’s triangle.

A ball that lands in slot $k$ (counting from 0) after $n$ rows turned right $k$ times. There are

\binom{n}{k}=\frac{n!}{k!\,(n-k)!}

such paths, each with probability $\left(\tfrac12\right)^n$ . Twelve rows make $2^{12}=4096$ paths; the middle slot has $\binom{12}{6}=924$ of them, about 22.6%.

02The binomial distribution

When each turn goes right with probability $p$ , the chance of slot $k$ is

P(k)=\binom{n}{k}p^k(1-p)^{n-k}

The average slot is $np$ and the standard deviation $\sqrt{np(1-p)}$ . With $p=\tfrac12$ and 24 rows that is 12 with a spread of about 2.45; tilted to $p=0.7$ the average moves to 16.8 and the spread is about 2.24, a little narrower, but still a bell.

The story on the stage uses fixed random seeds: with two thousand balls, every slot’s measured share is within two percentage points of the theory.

03Why always a bell: the central limit theorem

Count each right turn as $+1$ and each left as $0$ : the slot is a sum of $n$ independent small amounts. The central limit theorem says that a sum of many independent random amounts, whatever each looks like, is distributed close to the normal curve

\frac{1}{\sigma\sqrt{2\pi}}\,e^{-(x-\mu)^2/(2\sigma^2)}

A normal distribution has about 68% within one standard deviation of the mean. A 24-row Galton board has 69%, already close.

Heights, measurement errors and exam scores are often bell-shaped for the same reason: each is the sum of many small independent influences.

Questions

Why is a real Galton board never perfectly symmetric?

Real balls collide, bounce more than one place and meet pegs that are never perfectly aligned, so left and right are not exactly even and the turns are not fully independent. The central limit theorem only needs many roughly independent nudges, so the shape is still close to a bell.

Who invented it?

The British scientist Francis Galton used it in the 1870s to demonstrate the normal distribution; it is also called a bean machine or quincunx.

How many balls before the bell shows?

With a few dozen the columns are ragged; a few hundred show the outline; a few thousand sit almost on the curve. The deviations shrink roughly as $1/\sqrt N$: a hundred times more balls, ten times smaller relative error.

The syllabus side

LabBinomial distributionWhere does P(X = k) = C(n, k) pᵏ(1 − p)ⁿ⁻ᵏ come from, and why is the mean np?LabNormal distribution & 3σWhy does the share within one, two or three standard deviations come out the same for every normal curve, whatever μ and σ are?LabPermutations & combinationsWhy is the number of ways to choose k things from n equal to n(n−1)…(n−k+1) divided by k!, and why does Pascal’s triangle give the same numbers by adding?

Sources: Galton board — Wikipedia · Central limit theorem — Wikipedia · Pascal’s triangle — Wikipedia