Saramsha

No. 031  ·  17 October 2026

Every Peg Is a Coin Flip

A ball falls through rows of pegs, and at every one it has no way to choose: it goes left or right by pure chance, with no memory of the bounce before. Drop a thousand of them and the lopsided bell curve you get, every single time, isn't assumed — it falls straight out of the bouncing.

About this piece

No ball is rigged, and the curve still isn't fair

This board, usually called a Galton board or a bean machine after the Victorian statistician Francis Galton who built the first one, looks like it ought to spread balls evenly across the bottom. Every peg is a fair coin flip — left or right, fifty-fifty, no memory of the last bounce. If each bounce is unbiased, shouldn't every landing bin be equally likely? It feels like it should, and it is almost the only intuition this page exists to correct.

The bounces are fair; the paths are not evenly distributed among the bins. To land in the leftmost bin a ball must bounce left at every single peg, which is one exact sequence out of all the possible ones. To land in the middle bin, by contrast, a ball can bounce left-then-right, right-then-left, left-left-then-right-right in any order — any sequence with about as many lefts as rights gets you there, and there are vastly more such sequences than there are all-left or all-right ones. With n rows of pegs, the number of distinct paths into bin k (counting from the left, 0 to n) is the binomial coefficient "n choose k" — exactly Pascal's triangle, the one schools draw by adding adjacent numbers. Row 12 of that triangle runs 1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1: nearly a thousand times more paths lead to the centre than to either edge. That ratio, not any bias in the pegs, is the entire shape of the curve.

Why it ends up looking like a bell

As the number of rows grows, the jagged bar-chart of "number of paths per bin" smooths into the bell-shaped normal curve — a result old enough to predate Galton himself, proved in the 1730s by Abraham de Moivre and sharpened later by Laplace, long before anyone built a board to show it falling out of wood and glass. It is also the simplest working example of the central limit theorem: add up enough small, independent, equally-likely nudges — left or right, heads or tails, this error or that one — and their sum always arranges itself into the same bell shape, regardless of what the individual nudges were. That is the real reason the normal distribution shows up everywhere from exam scores to manufacturing tolerances to the noise in a measuring instrument: it is what you get whenever an outcome is the sum of many small independent pushes, which describes an enormous number of things in the world that have nothing to do with pegs.

Each ball's whole path is decided the instant it drops — one fair coin flip for every row of pegs, which is exactly the random draw the maths above describes — and its fall is then timed to real gravity, speeding up row by row the way a dropped object actually does, with a quick sideways kick at each peg it passes. What you see falling is the genuine result, played back at the speed gravity would give it, not a reenactment with a different answer underneath. Past about fifty balls in flight at once, or with reduced motion requested, new balls resolve the same draw instantly and add straight to the bin they land in, so dropping thousands costs no more per frame than dropping one. The dashed curve is the exact binomial prediction for the current row count — what the normal curve is approximating — scaled so a perfectly smooth run would just touch the centre bin.