Balls fall through a forest of pins. At every pin, a fair coin: left or right. No ball knows anything, no ball aims anywhere, no two balls take the same path, and yet the pile at the bottom builds the same shape every single time. Let it rain, and watch pure accident assemble the most famous curve in science.
For the curious kid.
Why the middle? To land far to one side, a ball needs almost every coin to say the same thing: twelve rights in a row happens once in four thousand balls. To land in the middle, the rights and lefts only need to roughly cancel, and there are thousands of different paths that do that. The board isn't guiding the balls; it is counting paths. The middle wins because the middle has the most ways to happen. That's the whole secret, and it is the same secret behind why people's heights bunch in the middle, and why careful measurements scatter the way they do: anything built from many small independent accidents piles up into this one curve. Try the bias slider (tilt every coin) and watch the pile lean while keeping its shape.
Deeper.
The board is a machine for computing Pascal's triangle: the number of paths to bin k after n pins is the binomial coefficient C(n,k), so the pile approaches the binomial distribution, drawn here as blue steps. In 1733 Abraham de Moivre, tired of adding enormous binomial sums for gambling problems, found a shortcut: for large n the whole staircase hugs one smooth curve, e−x², the first appearance of the bell curve in history, drawn here in gold. Slide the rows up and watch the staircase press itself against the curve.
The modern form of that observation is the central limit theorem: sum enough independent small pushes, whatever their individual quirks, and the total is distributed normally. It is one of mathematics' great universality results; the details of the pieces wash out and one shape remains. Francis Galton, who built this device in 1889 to study heredity, called the theorem the "supreme law of Unreason": wherever chaos is made of many small independent chances, order of exactly this shape appears. An honesty note, museum policy: the coins here are honest pseudo-random flips, checked against the exact binomial (every bin within ordinary statistical fluctuation), and the gold curve is de Moivre's normal approximation with mean np and variance np(1−p), no fitting to the data, ever. What you see agree, agrees.
No ball knows where it is going. The crowd of them has never once failed to arrive.