It's Just Chance

One flip is random. A thousand flips are a law.

A single coin flip is genuinely unpredictable — heads or tails, no way to call it. Yet the moment you repeat, structure appears. The share of heads lurches around at first, then quietly settles toward the true probability and stays there. Randomness doesn't disappear; it averages out.

Flip a fair coin a few times — the line whips around. Flip it a thousand times — it presses flat against 0.50. That convergence is the engine under every average you've ever trusted.

Average anything enough times and you get a bell.

Take the average of a sample — from any distribution, however lopsided. Collect many of those averages and they always shape themselves into the same curve: the normal bell. The underlying population can have two humps, a long tail, anything. The distribution of its means doesn't care.

The population strip below has two peaks — nothing bell-shaped about it. Yet the averages of n draws from it land in a clean bell. Raise n and the bell tightens. This is why so much of the world looks normal even when its parts don't.

A positive test usually means you're fine.

The test came back positive. Are you sick? Surprisingly often: probably not. When a disease is rare, the few true cases are buried under a pile of false alarms — because even a small error rate, applied to the huge healthy majority, produces more wrong positives than there are sick people. The intuition is almost impossible to feel from the formula. So count the dots.

1,000 people. Red = actually sick. The test splits them into negative and positive. Look at the positive column: at 1% prevalence with a good 90% test, most of those red-flagged dots are still blue — healthy people caught by a false alarm.

Slide prevalence up: as the disease gets common, the same test suddenly becomes trustworthy. The test never changed — only how rare the thing it's looking for is. That's Bayes: what you should believe after the evidence depends on what was true before it.