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Mathematics, Made Visual · No. 92 of the first 100

Probability

Drop a hundred balls of pure chance and watch them organise themselves into the most famous curve in science.

Plate XXXI — Order from accidents binomial → Gaussian
Drop 150 balls of pure chance and watch the bell curve assemble itself.
every peg: a fair coin flip, left or right0 balls
FIG. XXXI — ORDER FROM ACCIDENTS
Drop balls
Balls landed
0
Coin flips each
9pegs
No ball knows where it's going — at every peg it's a pure coin flip. One ball is unpredictable. But drop 150 and a perfect hill grows in the middle, every time. Middle bins are easy to reach (many paths lead there); the edges need nine flips in a row to agree. Order grows out of pure accident.
The short answer

On the Galton board, every ball bounces left or right at every peg by pure coin-flip — totally unpredictable. But drop lots of balls and a smooth hill always grows in the middle. Chance is unpredictable one ball at a time, and dead reliable in crowds.

What's actually happening

The Galton board is a machine for confessing what randomness really is. Watch one ball: nine coin flips, a drunken stagger, an unguessable destination. Watch a hundred and fifty: a smooth, symmetrical hill rises in the centre bins, every single time, with edges thinning out exactly on schedule. Nothing about ball #73 became predictable — but the crowd did. That is probability's whole bargain: it gives up on individuals and becomes precise about populations.

The hill's shape has a reason. To land in the middle bin, a ball needs roughly equal lefts and rights — and there are many different flip-sequences that do that (LRLRLRLRL, RRLLRLLRL…). To land at the far edge it needs all nine flips to agree, and there is exactly one sequence for that. The bins simply count routes: middle destinations are reachable by thousands of paths, extremes by one. Pile up the route-counts and you get the binomial distribution, which smooths — as rows increase — into the bell curve.

Here is why this one toy explains so much of the world: the bell curve appears wherever an outcome is the sum of many small, independent accidents. Your height (thousands of genes plus nutrition), a measurement's error (dozens of tiny disturbances), a poll's noise — each is a ball falling through its own pegboard. Francis Galton, who built the first board in 1874, called the effect "the supreme law of unreason": individually lawless events, collectively lawful. It's why casinos always profit and insurers can promise — they never know the next ball, and never need to.

Try it at home The streak hunt
  1. 1Flip a real coin 50 times, recording H/T. Separately, write down a fake "random" 50-flip sequence by hand, trying to make it look real.
  2. 2Now count the longest run of identical results in each. Hand-faked sequences almost never dare a run longer than 3; real chance usually produces a run of 5 or 6.
  3. 3Show both to a friend and ask which is real. Genuine randomness is streakier than anyone believes — which is also why hot streaks in games feel so meaningful and usually aren't.