The Coin Has No Memory

In August 1913, at the Casino de Monte-Carlo, black came up on the roulette wheel twenty-six times in a row. Gamblers crowded the table betting on red, reasoning that a correction was overdue, and each spin made the reasoning feel stronger. The house took millions of francs.

The wheel has no memory. It has no mechanism for having one. Each spin is a fresh physical event with the same probabilities as the first, and the sequence of previous results exists only in the minds of the people watching.

The intuition that something must even out is not baseless. It is a garbled version of a real theorem, and the garbling is precise enough to be worth locating.

The law of large numbers says that as the number of tosses grows, the *proportion* of heads approaches a half. It says nothing about the *difference* between the number of heads and the number of tails. That difference typically grows, not shrinks: after n tosses it is of order the square root of n. After a million tosses you expect the gap to be around a thousand.

Both statements are true at once, and they are compatible because a thousand out of a million is a tenth of a per cent. The proportion converges not because the gap closes but because the gap grows more slowly than the total. Nothing repairs an early run. It is diluted, and dilution is not correction.

So a hundred heads in a row does not make tails more likely next time. It should, however, make you wonder about the coin. If you have seen a hundred heads, the hypothesis that the coin is fair is in serious trouble, and the rational update is towards bias, which points in the opposite direction from the gambler's fallacy. This is why the fallacy is a fallacy and not merely an inefficiency: it gets the direction wrong.

Its mirror image causes as much trouble. Extreme results are followed by more ordinary ones, and this looks exactly like an effect.

Galton noticed it in the 1880s. Tall fathers have sons who are tall, on average, but less tall than their fathers. Short fathers have sons who are short, but less so. He called it regression towards mediocrity and initially thought it a biological force pulling the population back to type.

It is not a force. It is arithmetic. A father's exceptional height combines a heritable component with everything else: nutrition, chance, measurement. The heritable part is passed on; the rest is not, and there is no reason for a new roll of the dice to land in the same extreme place. So the son sits closer to average, and nothing has acted on anyone.

This manufactures evidence, continuously, for whatever you happened to do after an extreme result.

Kahneman tells the story of the Israeli flight instructors who were certain that praise made pilots worse and criticism made them better. They had observed it: praise an exceptional landing and the next is worse; criticise a terrible one and the next improves. They were observing regression to the mean, and they had built a training philosophy on it.

The pattern repeats everywhere. Treat the sickest patients and they improve somewhat. Intervene in the worst-performing schools and results tick up. Install cameras at the junctions with the worst accident records and casualties fall. In every case the units were selected because they had an unusually bad year, part of that was luck, and luck does not persist. Traffic engineers have a name for it (the regression-to-mean bias in blackspot schemes) and correct for it explicitly, because otherwise the evaluation measures the selection rule rather than the intervention.

The only reliable defence is a comparison group chosen the same way and left alone.

One more consequence, since it distorts a great deal of published ranking. Small samples produce extreme results more often, because with a small denominator one or two cases swing the rate. The counties with the highest cancer rates in the United States are small and rural; so are the counties with the lowest. No feature of rural life explains both. Any league table of small units by a rate is substantially a table of which units are smallest, and a good deal of school and hospital policy has been built on exactly that.