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 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.

In criticism of "biocentrism" Steven Novella describes this as the "lottery fallacy":

Deepak Chopra and Robert Lanza promote the notion of “biocentrism” – “that an accurate understanding of the world requires putting observers firmly into the equation, and that life may not be the accident of physics and chemistry that evolution suggests.” This idea is really nothing new – it is a transparent abuse and misunderstanding of modern physics and quantum mechanics in order to insert mysticism into science(...) 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 [won] by chance alone, because the odds were against it (...) So Chopra and Lanza get it backwards.[2]