# Bean machine

The **bean machine**, also known as the **quincunx** or **Galton box**, is a device invented by Sir Francis Galton to demonstrate the law of error and the normal distribution.

The machine consists of a vertical board with interleaved rows of pins. Balls are dropped from the top, and bounce left and right as they hit the pins. Eventually, they are collected into one-ball-wide bins at the bottom. The height of ball columns in the bins approximates a bell curve.

Overlaying Pascal's triangle onto the pins shows the number of different paths that can be taken to get to each pin.

A large-scale working model of this device can be seen at the Museum of Science, Boston.

## Distribution of the balls

If a ball bounces to the right *k* times on its way down (and to the left on the remaining pins) it ends up in the *k*th bin counting from the left. Denoting the number of rows of pins in a bean machine by *n* the number of paths to the *k*th bin on the bottom is given by the binomial coefficient . If the probability of bouncing right on a pin is *p* (which equals *0.5* on an unbiased machine) the probability that the ball ends up in the *k*th bin equals . This is the probability mass function of a binomial distribution.

According to the central limit theorem the binomial distribution approximates normal distribution provided that *n*, the number of rows of pins in the machine, is large.

## Games

Several games have been developed utilizing the idea of pins changing the route of balls or other objects: