The Galton Board (also called a bean machine or quincunx) was invented by Sir Francis Galton in 1876 to demonstrate how a normal distribution arises from many small, independent random events.
Balls are dropped from the top and hit a triangular grid of pins. At each pin, a ball bounces either left or right with equal probability — like flipping a coin. After passing through all rows, each ball lands in one of the collection bins at the bottom.
Each ball's final position is the result of many independent left/right decisions. A ball that goes right k times out of n rows lands in bin k. The number of paths leading to each bin follows the binomial coefficients (Pascal's triangle): there are many more paths to the center bins than to the edges.
For example, with 10 rows there is only 1 path to the far-left bin (all lefts), but 252 paths to the center bin — making it about 252 times more likely.
The Central Limit Theorem (CLT) is one of the most important results in statistics. It states that the sum (or average) of many independent random variables will approximate a normal (Gaussian) distribution, regardless of the shape of the original distribution.
The Galton Board is a physical demonstration: each ball's final position is the sum of n independent coin flips (+1 for right, 0 for left). As you increase the number of rows and balls, the histogram in the bins converges to a smooth bell curve — the normal distribution.
With n rows, the probability of landing in bin k is:
P(k) = C(n, k) · pk · (1−p)n−k
where C(n, k) = n! / (k! · (n−k)!) is the binomial coefficient and p is the probability of going right (0.5 for a fair board). Use the bias slider to explore how changing p skews the distribution. As n grows, the distribution converges to a normal curve with mean n · p and standard deviation √(n · p · (1−p)).