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 0.999¹⁰⁰⁰, 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 − 0.999¹⁰⁰⁰⁰ = 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.

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.

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).

...
Wikipedia