Law of truly large numbers
The law of truly large numbers (a statistical adage), attributed to Persi Diaconis and Frederick Mosteller, states that with a sample size large enough, any outrageous thing is likely to happen. Because we never find it notable when likely events occur, we highlight unlikely events and notice them more. The law seeks to debunk one element of supposed supernatural phenomenology. It is meant to make a statement about probabilities.
For a simplified example of the law, assume that a given event happens with a probability for its occurrence of 0.1%, within a single trial. Then, the probability that this so-called unlikely event does not happen (improbability) in a single trial is 99.9% (0.999).
In a sample of 1000 independent trials, however, the probability that the event does not happen in any of them, even once (improbability), is 0.9991000, or approximately 36.8%. Then, the probability that the event does happen, at least once, in 1000 trials is 1 − 0.9991000 ≈ 0.632 or 63.2%. This means that this "unlikely event" has a probability of 63.2% of happening if 1000 independent trials are conducted, or over 99.9% for 10,000 trials.
The probability that it happens at least once in 10,000 trials is 1 − 0.99910000 ≈ 0.99995 = 99.995%. In other words, a highly unlikely event, given enough trials with some fixed number of draws per trial, is even more likely to occur.
In criticism of pseudoscience
The law comes up in criticism of pseudoscience and is sometimes called the Jeane Dixon effect (see also Postdiction). It holds that the more predictions a psychic makes, the better the odds that one of them will "hit". Thus, if one comes true, the psychic expects us to forget the vast majority that did not happen (confirmation bias). Humans can be susceptible to this fallacy.
Another similar (to a small degree, see Psychologism and Anti-psychologism) manifestation of the law can be found in gambling, where gamblers tend to remember their wins and forget their losses, even if the latter far outnumbers the former (though depending on a particular person's environment, behaviors, customs or habits, so the opposite may also be local truth – statistical prevalence not featured). Mikal Aasved links it with "selective memory bias", allowing gamblers to mentally distance themselves from the consequences of their gambling by holding an inflated view of their real winnings (or losses in the opposite case).
- Infinite monkey theorem
- Large numbers
- Law of large numbers
- Law of small numbers
- Littlewood's law
- Look-elsewhere effect
- Black swan theory
- Murphy's Law
- Psychic phenomena
- The Library of Babel
- Everitt 2002
- 1980, Austin Society to Oppose Pseudoscience (ASTOP) distributed by ICSA (former American Family Foundation) "Pseudoscience Fact Sheets, ASTOP: Psychic Detectives"
- Daniel Freeman, Jason Freeman, 2009, London, "Know Your Mind: Everyday Emotional and Psychological Problems and How to Overcome Them" p. 41
- Mikal Aasved, 2002, Illinois, The Psychodynamics and Psychology of Gambling: The Gambler's Mind vol. I, p. 129
- Weisstein, Eric W. "Law of truly large numbers". MathWorld.
- Diaconis, P.; Mosteller, F. (1989). "Methods of Studying Coincidences" (PDF). Journal of the American Statistical Association. American Statistical Association. 84 (408): 853–61. doi:10.2307/2290058. JSTOR 2290058. MR 1134485. Archived from the original (PDF) on 2010-07-12. Retrieved 2009-04-28.
- Everitt, B.S. (2002). Cambridge Dictionary of Statistics (2nd ed.). ISBN 052181099X.
- David J. Hand, (2014), The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day