The story of the Galton board

A Galton board is a board with staggered rows of pegs. You drop balls from the top, and they collect in bins at the bottom. At every peg a ball goes either left or right. Where one ball ends up is impossible to predict. But drop many, and they settle into the same shape every time: high in the middle, low at the sides, like a bell.

Why a bell shape?

Suppose that at each row a ball is equally likely to go left or right. With 8 rows there are 2 to the 8th, which is 256, different paths. To land in the outermost bin a ball must turn the same way 8 times, so there is only one such path. But to land in the middle bin it needs 4 rights and 4 lefts in any order, which gives 70 paths. The path counts for the bins from one edge to the other are 1, 8, 28, 56, 70, 56, 28, 8, 1. These are the numbers in row 8 of Pascal's triangle.

This site's program enumerated every path for 1 to 12 rows and confirmed that each bin's probability equals its number of paths divided by the total number of paths. Mathematics also tells us that with many rows and many balls, this shape approaches the smooth bell curve called the normal distribution.

Galton and his device

The device was devised by Francis Galton in England. It is said to have been designed in the 1870s and to have become widely known when his 1889 book Natural Inheritance described it with a diagram. Sources disagree on the exact year. Its original name is the “quincunx”, an old word for a pattern of five dots. It is also called a “bean machine”.

Galton was struck by how a pile-up of unrelated chances gives rise to a regular pattern. His book has a section on order in apparent chaos.

One more thing should be said. Although Galton contributed greatly to statistics, he also promoted eugenics, an idea that is rejected today as wrong. The mathematics of the Galton board has nothing to do with it, and the board is still used to show how probability works.

How to think about it

• Count only the rights. The order of turns doesn't matter. The number of right turns decides the bin.
• Count paths. Few paths lead to the edges and many lead to the middle, which is why balls pile up in the middle.
• Add up probabilities. Multiply the probabilities along each path, then add the paths that reach the same bin to get that bin's probability.
• Compare different sizes. More rows make the shape smoother, and more balls make the real pile match the theory.

The board on this site is a simplified probability model: it does not calculate collisions or speed. On the puzzle page you can change the rows, the number of balls and the chance of going right, and compare the bars with the dashed theory line.