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Random Balls Build a Perfect Bell Curve
Thousands of balls, every bounce a coin flip — and a perfect bell curve appears. Headphones on.
Focus with this filmEvery ball goes left or right at random. Thousands of them still draw the same bell curve.
All about the galton board →4 chapters
- 0:00 Every ball flips a coin at every peg
- 0:10 The curve is drawn first — watch the balls fill it
- 0:47 Rig the pegs: 65% of bounces go right
- 1:34 Twice the rows, a sharper bell
The math
At each peg a ball goes left or right with equal chance, independently of everything before. After 12 rows its bin is simply the number of times it went right — 12 coin flips. There is exactly one route to the far-left bin (left, left, left… twelve times) out of 4,096 possible routes, but 924 different routes to the centre bin (any six lefts and six rights). More routes means more balls, and the route counts are the numbers of Pascal's triangle: 1, 12, 66, 220, 495, 792, 924… Plotted, they form the binomial distribution, which for many flips becomes the normal "bell" curve — de Moivre found this in 1733, and Francis Galton built this board in 1876 to show it. In the second act every peg is rigged to send 65% of balls right: the bell does not break, it just moves to 12 × 0.65 = 7.8. In the third act each ball flips 24 coins instead of 12: relative to the board, the bell gets narrower, because the spread grows only with the square root of the number of flips. The white curve is drawn before the balls arrive; every ball's route is a genuine random draw, and the run shown is one take chosen for how cleanly it fills the curve.
Colour is route: balls that went left more than expected are teal, right more than expected pink, right on target white.
