Galton board

Distribution of the beads

If a bead bounces to the right k times on its way down (and to the left on the remaining pegs) it ends up in the kth bin counting from the left. Denoting the number of rows of pegs in a Galton Board by n, the number of paths to the kth bin on the bottom is given by the binomial coefficient ${\displaystyle (\genfrac{}{}{0ex}{}{n}{k})}$. Note that the leftmost bin is the 0-bin, next to it is the 1-bin, etc. and the furthest one to the right is the n-bin - making thus the total number of bins equal to n+1 (each row does not need to have more pegs than the number that identifies the row itself, e.g. the first row has 1 peg, the second 2 pegs, until the n-th row that has n pegs which correspond to the n+1 bins). If the probability of bouncing right on a peg is p (which equals 0.5 on an unbiased level machine) the probability that the ball ends up in the kth bin equals ${\displaystyle (\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}}$. This is the probability mass function of a binomial distribution. The number of rows correspond to the size of a binomial distribution in number of trials, while the probability p of each pin is the binomial's p.

According to the central limit theorem (more specifically, the de Moivre–Laplace theorem), the binomial distribution approximates the normal distribution provided that the number of rows and the number of balls are both large. Varying the rows will result in different standard deviations or widths of the bell-shaped curve or the normal distribution in the bins.

Another interpretation more accurate from the physical view is given by the entropy: since the energy that is carried by every falling bead is finite, so even that on any tip their collisions are chaotic because the derivative is undefined (there is no way to previously figure out for which side is going to fall), the mean and variance of each bean is restricted to be finite (they will never bound out of the box), and the Gaussian shape arises because it is the maximum entropy probability distribution for a continuous process with defined mean and variance. The rise of the normal distribution could be interpreted as that all possible information carried by each bean related to which path it has travelled has been already completely lost through their downhill collisions.

Examples

History

Francis Galton designed his board as part of a presentation for the Royal Institution Discourses on February 27, 1874. His goal was to promote the use of ranking instead of measurement in statistics, so that qualities such as intelligence could be assigned numbers without requiring experimental data. The piling of the balls into a normal distribution was supposed to illustrate how a mean value would emerge from multiple tests.

Francis Galton wrote in 1889 his book Natural Inheritance:

Order in Apparent Chaos: I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the Law of Frequency of Error. The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.

However, Galton also wished to demonstrate that extreme values of intelligence result from heredity, in apparent contradiction with his experiment since it produces extreme values as dispersion from randomness alone. Aware of this issue, he tried to address it in 1875 by arguing that his box did not reflect situations where bias would be introduced by what he called a main influence factor.

In a 1877 letter to George Darwin, Galton described a second version of the board, with two stages, where the compartments at the bottom of the highest stage had small trapdoors that would allow the balls of one chosen compartment to fall through the second stage. His goal was to illustrate his concept of "reversion to mediocrity", i.e. that without marriage control the "best" parts of the population would mix with the "mediocre", so that their offspring would gradually revert towards an average value. This version, however, was not built.

Games

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

• Bagatelle
• Pachinko
• Payazzo
• Peggle
• Pinball
• Plinko
• The Wall

It is suggested that bagatelle provided inspiration for Galton's device.