The Galton Board: A Walk to Normality

The Galton Board: A Walk to Normality

The Galton Board, or quincunx, was invented by Sir Francis Galton to visually demonstrate the central limit theorem. By dropping marbles through a grid of pegs, the board shows how multiple random events create a predictable, well-defined distribution.

Mathematical Path Explanation

When a marble hits a peg, it has an equal probability of moving left or right. We assume the probability of going right $p$ is $0.5$, and left $1 - p$ is also $0.5$. If the board has $n$ rows, each marble makes $n$ random decisions. Its final position depends on the total number of rightward moves.

Relation to Binomial Distribution

The distribution of marbles in the bins follows a binomial distribution. This describes the number of successes ($k$) in a sequence of $n$ independent trials. On the Galton Board, $n$ is the number of rows, and $k$ is the number of right turns. The probability of a marble landing in a specific bin is given by the binomial probability mass function:

From Binomial to the Bell Curve

As the number of rows ($n$) and marbles increases, the binomial distribution increasingly approximates the normal distribution, also known as the Gaussian bell curve. This is the key principle the Galton Board demonstrates: the sum of many independent random variables tends toward a normal distribution, described by the probability density function:

On our board, the mean $\mu$ is $np$ and the variance $\sigma^2$ is $np(1-p)$, corresponding to the binomial distribution. This elegantly shows how the predictable order of the bell curve emerges from the randomness of individual events.