ObservedBinomial expectationNormal approximation
THE STATISTICS
Unpredictable paths. Predictable pattern.
Each peg represents one Bernoulli trial: the ball moves right with probability p and left with probability 1 − p. A single route is uncertain, but the totals from many balls settle around the binomial distribution.
P(X=k) = C(n,k) pk(1−p)n−kprobability of landing in bin k
Small experiments fluctuate. As more balls fall, the observed proportions usually move closer to their theoretical values.
At p = 0.5 the distribution is symmetric. Changing p shifts both the mean and the most likely landing bins.
The board is associated with Francis Galton. His statistical work was influential, but his advocacy of eugenics was harmful and remains ethically discredited.