# 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.^{[1]} 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.

## Example

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 0.999^{1000}, 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 1 - 0.999^{10000} = 0.99995 = 99.995%.

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. 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.^{[2]} Humans can be susceptible to this fallacy.

A similar (to small degree, see: psychologism vs. anti-psychologism) manifestation can be found in gambling, where gamblers tend to remember their wins and forget their losses^{[3]} (but depending on particular person individual environmental behaviors, customs or habits: so the opposite is also local truth^{[4]} - *statistical prevalence not featured*) and thus hold an inflated view of their real winnings (or losses respectively). Aasved links it with "selective memory"^{[4]} (synonymous: kinds of "amnesia").

## See also

- Coincidence
- Large numbers
- Law of large numbers
- Law of small numbers
- Littlewood's law
- Look-elsewhere effect
- Miracle
- Murphy's Law
- Psychic phenomena
- Infinite monkey theorem

## Notes

- ↑ 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 - ↑
^{4.0}^{4.1}Mikal Aasved, 2002, Ilinois,*"THE PSYCHODYNAMICS AND PSYCHOLOGY OF GAMBLING: The Gambler's Mind"*vol. I, p. 129

## References

- 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): 853–861. JSTOR 2290058. MR 1134485. doi:10.2307/2290058. 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