🔔 The Binomial Distribution and the Galton Board

About this simulation

Invented by Francis Galton in the 1880s, a Galton board (also called a quincunx) drops beads through staggered rows of pegs. At every peg a bead bounces left or right with roughly equal chance, exactly like a coin flip. After n rows the bead has made n independent left/right "trials," and the bin it lands in records how many of those trials went right.

How it works

• Each peg encounter is a Bernoulli trial: success (right) with probability p, failure (left) with probability 1−p.
• The bin index a bead lands in is the count of "successes" out of n rows — a sample from the Binomial(n, p) distribution.
• As thousands of beads accumulate, the bar heights converge to the binomial probability mass function, shown as the amber curve overlay.
• The distribution's mean is n·p and its standard deviation is √(n·p·(1−p)) — for large n it looks increasingly like a bell-shaped normal curve, a visual proof of the Central Limit Theorem.

Controls

• Rows of pegs (n) — number of Bernoulli trials each bead undergoes (3–14).
• Right probability p — bias of each bounce; 0.50 is a fair "coin," away from 0.50 skews the histogram.
• Drop rate — how many beads are released per second.
• Show binomial curve overlay — draws the theoretical PMF scaled to the current bead count over the bars.
• ↺ Clear bins — empties all bins and restarts the count. Drag to rotate, scroll to zoom.

Did you know?

Galton originally built this device to argue that heredity — traits like height — follows the same bell curve produced by summing many small independent random influences, laying groundwork for regression analysis and modern statistics.