Galton Board: Order Hidden in Chaos (Video)

When we see the Galton board in action, our first thought is that its inventor, a statistician and anthropologist, was lucky not to live in the Middle Ages. The simple fact that all the little balls fall following a curve every time the device is turned almost seems like an act of sorcery, but in reality it's pure central limit theorem. However, that's not all: the machine becomes even more interesting when you place a copy of Pascal's triangle on it, because from that combination come some very useful resources.

We turn, and a curve forms. Another turn, and a curve again. No matter how many times we try, the result is (more or less) the same: thousands of balls creating a curve at the bottom. The Galton board, designed by Sir Francis Galton, serves as a demonstration of the normal distribution and the central limit theorem.

In very relaxed terms, that theorem establishes that when a very large number of samples are taken from a population, the distribution of the average of the samples will approach a normal distribution. It's undoubtedly one of the most profound and essential concepts in the world of statistics… and it melts brains easily. But let's get back to the machine.

The Galton Board

The posts of the Galton board are positioned like a pyramid. When a ball hits a post, it has a 50 percent probability of going left or right. When passing to the second row of posts, that process repeats, keeping 50 percent.

Through a large number of attempts, the expected result is that those balls with a very similar number of left and right movements are the most common. In other words, the number of balls in the center will be greater.

The probability that a ball falls into one of the slots at the bottom follows the concept of binomial distribution, but with the intervention of the aforementioned theorem, the binomial distribution approximates a normal distribution.

The Galton board gains an extra layer of complexity when placing a representation of Pascal's triangle over the posts. In essence, the number of each cell in the triangle tells us the number of paths a ball can take to reach that post.

The numbers in the triangle are larger in the center, meaning a greater number of available routes for the balls. Look at this copy of the triangle, and the bottom cell in the center. A ball has 12,870 options to hit there.