Thought Toys · Exhibit 05

The Galton board

Each bead clatters down through the pins on a path no one could call — a coin-flip at every step. Drop one and it's pure chance where it lands. Drop a few thousand and the same blind bouncing stacks them, every single time, into the one shape the math drew before the first bead fell.

0Beads dropped
—Mean bin
—Std dev

No beads dropped yet. An amber bell curve marks where the math predicts the beads will pile up.

falling bead where they landed the predicted bell curve
fewmany
lean leftlean right
slowfast
Bins always total to the beads dropped.

What you're seeing

Follow a single bead. At the first pin it goes left or right — a coin toss. At the next pin, another toss. By the time it reaches the bottom it has flipped a coin a dozen times, and where it lands is just the running tally of those flips. You genuinely cannot predict one bead. The path is a tiny, private streak of luck.

Now press Rain and stop watching any one bead. A bead that drifts far left needed an unlikely run of left-bounces; almost all of them, by sheer arithmetic, end up somewhere near the middle, where the lefts and rights roughly cancel. The far bins are starved and the centre is fed — so the pile grows tall in the middle and tapers to the edges. The amber curve was drawn first, before a single bead fell; the beads simply find their way into it. Wild one at a time, dead reliable by the thousand.

The bell isn't a coincidence of this toy. Drag Pin bias and the whole heap slides to one side, still bell-shaped, just re-centred. Add rows and it grows narrower in proportion and smoother. Anything that's a sum of many small independent nudges — measurement errors, heights in a crowd, the noise on a signal — piles up the same way. That's why the bell curve is everywhere: it's the shape randomness makes when you let it add up.

The rule, exactly. A bead meets n rows of pins and at each one goes right with probability p (left otherwise), independently. It lands in bin k after exactly k right-bounces — so the bins follow the binomial distribution, P(k) = C(n,k)·pk(1−p)n−k, with mean n·p and standard deviation √(n·p·(1−p)). For p = ½ it's symmetric and centred on n/2. As n grows the binomial is hugged ever more tightly by the normal “bell” curve of the same mean and spread — de Moivre–Laplace, 1733, the original central-limit theorem. The amber curve is that normal, scaled to the beads dropped. (Checked offline: 400k-bead simulations match the pmf to ~0.001, and the running mean and spread land on n·p and √(n·p(1−p)).)

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