Bean machine

The bean machine, also known as the or Galton box, is a device invented by Sir Francis Galton to demonstrate the central limit theorem, in particular that the normal distribution is approximate to the binomial distribution. Among its applications, it afforded insight into regression to the mean or "regression to mediocrity".

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

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

Large-scale working models of this device can be seen in the Mathematica: A World of Numbers... and Beyond exhibits permanently on view at the Boston Museum of Science or the New York Hall of Science.

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 kth bin counting from the left. Denoting the number of rows of pins in a bean machine by n, the number of paths to the kth bin on the bottom is given by the binomial coefficient ${\displaystyle {n \choose k}}$. 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 kth bin equals ${\displaystyle {n \choose k}p^{k}(1-p)^{n-k}}$. This is the probability mass function of a binomial distribution.

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