Law of truly large numbers
The law of truly large numbers, 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.
For a simplified example of the law, assume that a given event happens with a probability of 0.1% in one trial. Then the probability that this unlikely event does not happen in a single trial is 99.9% = 0.999.
In a sample of 1000 independent trials, the probability that the event does not happen in any of them is , or 36.8%. The probability that the event happens at least once in 1000 trials is then 1 − 0.368 = 0.632 or 63.2%. The probability that it happens at least once in 10,000 trials is .
This means that this "unlikely event" has a probability of 63.2% of happening if 1000 chances are given, or over 99.9% for 10,000 chances. In other words, a highly unlikely event, given enough tries, 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.
Humans can be susceptible to this fallacy. A similar manifestation can be found in gambling, where gamblers tend to remember their wins and forget their losses and thus hold an inflated view of their real winnings.
Steven Novella describes this as the "lottery fallacy":
It is also the lottery fallacy. If we hold a world-wide lottery and only one human in the 6.5 billion wins, the odds of that person winning is very small. But someone had to win. Chopra and Lanza are arguing that the winner could not have one [sic] by chance alone, because the odds were against it.
- Large numbers
- Law of large numbers
- Law of small numbers
- Littlewood's law
- Psychic phenomena
- Infinite monkey theorem
- Weisstein, Eric W., "Law of truly large numbers", MathWorld.
- Diaconis, P.; Mosteller, F. (1989). "Methods of Studying Coincidences". Journal of the American Statistical Association (American Statistical Association) 84 (408): 853–861. doi:10.2307/2290058. JSTOR 2290058. MR 1134485. Retrieved 2009-04-28.
- Everitt, B.S. (2002). Cambridge Dictionary of Statistics (2nd ed.). ISBN 052181099X.