The coin that remembers

A fair coin has landed heads six times in a row. What is the chance the next throw is heads?

Most people feel strongly that it is less than half. The coin, they reason, is due a tail. This feeling has a name — the gambler's fallacy — and the answer is that the chance is exactly half, because a coin has no memory and the previous throws have left no trace in it.

The mistake comes from a correct idea applied in the wrong place. Over a very large number of throws, the proportion of heads does approach half. People translate this into a belief that the coin corrects itself, as though a run of heads must be balanced by a run of tails. What actually happens is subtler: the run is not cancelled, it is diluted. Six extra heads matter enormously in twelve throws and are invisible in a million.

The practical version appears away from coins. A student who has had three questions wrong in a row is not more likely to get the fourth right. A bus that has been late four days running is not more likely to be on time today, unless something about the bus has changed. In both cases the feeling of being owed is strong and the mechanism that would deliver is absent.

The one exception matters. If the process can be affected by what has happened — a tired examiner, a driver under pressure after complaints — then history does change the odds. The question is always whether there is a mechanism, not whether there is a pattern.