The gambler's fallacy and independent trials
Posted 19 December 2006 at 22:40
A sequence of independent trials has no memory. The belief that it does is old enough and stubborn enough to have earned a name of its own, and the name is a little unfair, because the pattern of thinking behind it appears everywhere and not only at a table.
The formal statement is short. Two events are independent when the probability of the second, given that the first occurred, equals the probability of the second on its own. For a wheel, a die or a machine cycle, that condition holds by construction: nothing physical carries information from one resolution to the next. So the probability of an outcome after a run of ten contrary outcomes is exactly what it was before the run started.
The fallacy takes two forms and they contradict each other, which is a useful diagnostic. One form says a run makes the contrary outcome more likely, because the sequence is due to even out. The other says a run makes the same outcome more likely, because the sequence is running hot. Both cannot be right, and in fact neither is, because the sequence is not doing anything at all except producing independent draws.
The intuition behind the first form is not entirely groundless, which is why it survives. It is a garbled version of a true statement about long-run proportions: as the number of trials grows, the proportion of each outcome does approach its probability. What the garbled version adds, incorrectly, is a mechanism. The proportion approaches its value because later trials dilute the early imbalance, not because later trials correct it. The absolute imbalance is free to grow while the proportion shrinks.
Once that distinction is clear, most of the recording apparatus that surrounds these games becomes legible. Boards that display recent results are displaying a sequence that carries no information about the next result. They are not deceptive in any narrow sense, since they show exactly what happened. They simply invite a reading that the mathematics does not support.
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