Galton Board Simulator

The Galton Board (also called a bean machine or quincunx) was invented by Sir Francis Galton in 1889 to demonstrate how random processes naturally produce the bell-shaped normal distribution. Balls are dropped from the top and fall through an array of pegs. At each peg, a ball has a 50/50 chance of bouncing left or right — just like flipping a coin.

After passing through many rows of pegs, the balls collect in bins at the bottom. With enough balls, the height of each bin’s pile traces out a binomial distribution — which, by the Central Limit Theorem, converges to the Normal (Gaussian) bell curve as the number of rows grows.

The math in one line: The probability a ball lands in bin k after n peg rows is P(k) = C(n,k) × (1/2)n, where C(n,k) is “n choose k.” The gold curve overlay shows this theoretical binomial PMF scaled to the live data.

AP Statistics connection: This IS the bell curve in motion. Each ball is a sequence of independent coin flips. The bin it lands in equals the number of right-bounces out of n flips. The sum of many independent random variables converges to Normal — and you can watch it happen in real time.

Default layout — Rectangle (quincunx): The simulator starts in Rectangle mode with Bell Curve Priority enabled. This is the best setting for demonstrating the CLT: rows alternate between n and n+1 pegs, filling the full board width, and edge pegs sit close to the walls to prevent balls from escaping the edges. Result: a clean, textbook-quality bell curve. Switch to Triangle mode to see Galton’s original design — where the classic edge-leakage effect is visible and itself a teaching moment.  •  Rectangle edge modes: Bell Curve Priority (default in Rectangle): edge pegs sit close to the board walls, blocking edge escape and producing a cleaner distribution — but balls may occasionally get pinched between a peg and the wall and require a Reset to clear. Anti-Stuck: edge pegs inset far enough that a ball can always pass between peg and wall — no sticking, but some balls escape the edge, inflating the outermost bins. Choose Bell Curve Priority when demonstrating the CLT to students; use Anti-Stuck for uninterrupted long runs.  •  Triangle layout note: In Triangle mode, a small gap exists between the outermost pegs and the board walls — the same edge effect seen in physical Galton boards. This means the outermost bins often receive more balls than the theoretical model predicts: an honest representation of the physical device.  •  Ball–Ball Physics: When the Ball–Ball Physics toggle is on, balls exchange momentum on collision (useful for physics demonstrations), but this breaks the statistical independence assumption and the distribution will deviate from the theoretical binomial/Normal model. See the Chat Log for every design decision.