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>Home Page >Behavioral Finance >Law of Small Numbers (also Gamblers' Fallacy):
The 'Law of Small Number' is the fact that investors have a tendency to infer probabilities from a small series of recent historical data
Belief in the Law of Small Numbers, Tversky and Kahneman, 1971
"People have erroneous intuitions about the laws of chance. In particular, they regard a sample randomly drawn from a population as highly representative".
Inference by Believers in the Law of Small Numbers, Rabin, 2000
"A person exaggerates the likelihood that a short sequence of i.i.d. signals resembles the long-run rate at which those signals are generated. The model predicts that people may pay for financial advice from 'experts' whose expertise is entirely illusory. Other economic applications are discussed".
[RYAN] D-Day invasion at Normandy, 1944. "All along the chain of German command the continuing bad weather acted like a tranquilizer. The various headquarters were quite confident that there would be no attack in the immediate future. Their reasoning was based on carefully assessed weather evaluations that had been made of the Allied landings in North Africa, Italy and Sicily. In each case conditions had varied, but (German) meteorologists had noted that the Allies had never attempted a landing unless the prospects of favorable weather were almost certain. To the methodical German mind there was no deviation from this rule; the weather had to be just right or the Allies wouldn't attack. And the weather wasn't just right".
[BEN-ZVI] Operation Barbarossa, the German Invasion of Russia, 1941. "One example of an assumption of strategic possibility is reflected in Stalin's brief that Hitler must issue an ultimatum before was would break out. The fact that prior to April 8, 1941, Germany had made ultimate demands before undertaking military action convinced Stalin that this pattern would continue in the future". The sample size on which this was based was less than five.
[Wohlstetter] The Cuban Missile Crisis, 1962. "...(a failure of intelligence evaluation) was the predisposition of the intelligence community to the philosophical conviction that it would be incompatible with Soviet policy. Khrushchev had never put medium or long-range missiles in any satellite country and, therefore, it was reasoned, he certainly would not put them on an island 8,000 miles away from the Soviet Union, and only 90 miles away from the United States, when this was bound to provoke a sharp American reaction". The sample size on which this was based was less then five.
The `Conjunction Fallacy' Revisited: How Intelligent Inferences Look Like Reasoning Errors, HERTWIG and GIGERENZER, 1999
People's apparent failures to reason probabilistically in experimental contexts have raised serious concerns about our ability to reason rationally in real-world environments. One of the most celebrated of these failures is the conjunction fallacy, in which people violate what is widely considered the 'simplest and most fundamental qualitative law of probability' (Tversky and Kahneman), the conjunction rule. Stich, for instance, saw major implications of the conjunction fallacy for people's assessment of technological risks: 'It is disquieting to speculate on how large an impact this inferential failing may have on people's assessments of the chance of such catastrophes as nuclear reactor failures'. Kanwisher argued that the conjunction fallacy might underlie 'flawed arguments' that 'often recur in debates on U.S. security policy... Strategic priorities have in the past become distorted by overemphasizing the most extreme scenarios at the expense of less flashy but more likely ones'. Finally, based on the conjunction fallacy, Gould concluded more generally that 'our minds are not built (for whatever reason) to work by the rules of probability'.
The "Hot Hand" syndrome
"It is no great wonder if in the long process of time, while fortune takes her course, hither and thither... numerous coincidences should spontaneously occur"