Experiments…the lean mean bean machine

Statistics 101 taught us that the normal distribution can be used to approximate the binomial distribution. You too can investigate this hypothesis with the use of a simple piece of apparatus called the Galton board (named after the British scientist Sir Francis Galton), also known as a quincunx or bean machine.

The bean machine consists of a vertical board with interleaved rows of pins. Balls are dropped from the top, and bounce either left (with some probability p) or right (with probability q = 1 – p) as they hit the pins. Eventually, they are collected into one-ball-wide bins at the bottom.

For symmetrically placed nails, balls will bounce left or right with equal probability, so p=q=1/2. If the rows are numbered from 0 to N-1, the path of each falling ball is a Bernoulli trial consisting of N steps. Each ball crosses the bottom row hitting the nth peg from the left (where 0<=n<=N-1) if it has taken exactly n right turns, which occurs with probability.

If the number of balls is sufficiently large, then, according to the weak law of large numbers, the height of ball columns accumulated in the bins will eventually approximate a bell curve.

Want to see the experiment in action but not quite ready to invest in your own bean machine?  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 Museum of Science in Boston or the New York Hall of Science…or just play some good old fashioned pinball for a close (for want of a better word) approximation to the experience.

You can read more Lighter Side articles here.

Pamela Hellig