In random sequences, a streak of one outcome makes the opposite outcome more likely; the 'law of averages' must balance things out.
Independent random events have no memory. A fair coin's probability of landing heads is exactly 50% regardless of any preceding streak. The law of large numbers describes proportions converging over enormous sample sizes but makes no promise about any individual event or short sequence.
What changed?
On the evening of August 18, 1913, at the Casino de Monte-Carlo, a roulette ball landed on black. The run continued through fifteen consecutive blacks, then twenty, then twenty-six before a red finally appeared. With each additional black result, the gamblers surrounding the table increased their wagers on red, convinced the sequence had to reverse. Reports from that evening indicate that what was lost was substantial, and that the losses came from confidence in a principle that probability theory had never supported.
A roulette wheel has no memory of its preceding outcomes. The probability of red on any given spin is approximately 18/37, the same whether the previous result was red or whether the previous twenty-six were black. Nothing about the preceding sequence alters the mechanics of the next spin, and nothing in probability theory suggests it should.
Jacob Bernoulli established the mathematical framework in Ars Conjectandi, published posthumously in 1713. His law of large numbers proved rigorously that as the number of independent trials increases, the observed frequency of an outcome converges toward its theoretical probability. A coin flipped one million times will produce heads close to 500,000 times. The theorem is about proportions at enormous scale, and it is correct. The error arises when this long-run convergence is misread as a short-run correction: that the coin will produce heads more often now to compensate for a recent deficit. Bernoulli's theorem says nothing of the kind. The proportion approaches 50% not because the coin generates corrective heads but because any deviation becomes negligible against a growing denominator.
The confusion had a linguistic dimension. The phrase "law of averages" circulated in everyday speech and in some classroom settings as a looser version of Bernoulli's theorem, shorn of the mathematical precision that makes the theorem true. Where Bernoulli spoke of limiting proportions across an unlimited number of trials, the popular version promised balance across any sequence of reasonable length. The two ideas used overlapping language and pointed to opposite conclusions about any individual event.
Daniel Kahneman and Amos Tversky named and studied the error systematically in 1971, in a paper titled "Belief in the Law of Small Numbers" published in Psychological Bulletin. Their research showed that people consistently expected small random samples to match the anticipated long-run distribution: a sequence of coin flips reading HHHTTT felt more representative of a fair coin than HHHHHH, even though both are equally probable. The intuition that random sequences should look balanced at every scale, not merely in aggregate, was robust and resisted correction even among subjects with statistical training. Kahneman and Tversky classified it as a heuristic error, a systematic deviation from probability reasoning that presents itself as common sense.
The classroom version had a specific mechanism. Teachers introducing long-run frequency illustrated convergence with coin-flip diagrams, correctly showing that proportions balanced across thousands of trials. Students observed the pattern and drew an inference the mathematics did not support: that past imbalance generated pressure the future would relieve. The distinction between "balances eventually across a very large sample" and "will correct after this streak" was not always addressed explicitly, and the two were often conflated.
Research published after Kahneman and Tversky's paper found the fallacy operating in contexts far removed from gambling. Studies of parole board decisions showed that favorable rulings were less likely after a string of prior approvals, and unfavorable ones less likely after a string of prior denials, as if the boards were implicitly managing a quota. Studies of loan officers and of baseball umpires calling balls and strikes showed comparable patterns. The cognitive error appeared wherever people read sequences of outcomes and expected the sequence to self-correct.
