The Galton board The Galton board (or Quincunx) was devised by Sir Francis Galton to physically demonstrate the relationship between the binomial and normal distributions.

The Galton box consists of pegs arranged in a triangular pattern. Balls (or beans) are dropped from the top of the board, bounce among the pegs, and collect in bins at the bottom.

FAILURE SUCCESS At each peg the ball can fall left or right. If we consider a left turn to be a FAILURE and a right turn to be a SUCCESS we can see each pin as a Bernoulli experiment.

Each row in the Galton board represents an independent Bernoulli trial Trial 1 Trial 2 Trial 3 The board below represents a Binomial experiment with 3 trials.

0123 The bins represent the number of times that the ball moves to the left (failure) or right (success) at each level. The number of times we get 0 right turns (0 “successes”) The number of times we get 1 right turns (1 “successes”) The number of times we get 3 right turns (3 “successes”) The number of times we get 2 right turns (2 “successes”)

Suppose that the probability that a ball moves to the right when it hits a peg is p= The paths that the ball can follow to land in bin “1” are: {R,L,L}, {L,L,R}, or {L,R,L}. The probability of one of these paths is equal to: p ⨯ (1-p)⨯ (1-p) =0.125 p ⨯ (1-p)⨯ (1-p) = unique paths i.e., 3 unique paths. The probability that the ball lands in bin “1” is then: ⨯ 0.125=0.375 R L L

The probability that a ball lands in bin x is thus given by the binomial mass function (with n trials and probability of success equal to p ): The combination term represents the number of unique paths that the ball can follow.

Pascal’s Triangle Pascal’s triangle can also be used to determine the number of unique paths that the ball can follow. This is the number of unique paths for each bin in our example.

What is the relationship between the Normal distribution and the Binomial distribution? A well known property is that if the number of trials increases to infinity, then the Binomial distribution approximates the Normal distribution. This approximation is reasonable even for a small number of trials.

The Normal distribution result is due to the Central Limit Theorem. The Central Limit Theorem states that the sum of n independent identically distributed random variables is approximately Normally distributed as n becomes large.

The following animation can illustrate this approximation for various numbers of trials. Note the shape of the frequency bar plot constructed at the bottom of the animation.

Fin