Law of truly large numbers

Short description: Law of statistics

The law of truly large numbers (a statistical adage), attributed to Persi Diaconis and Frederick Mosteller, states that with a large enough number of independent samples, any highly implausible (i.e. unlikely in any single sample, but with constant probability strictly greater than 0 in any sample) result is likely to be observed. Because we never find it notable when likely events occur, we highlight unlikely events and notice them more. The law is often used to falsify different pseudo-scientific claims; as such, it is sometimes criticized by fringe scientists.

The law is meant to make a statement about probabilities and statistical significance: in large enough masses of statistical data, even minuscule fluctuations attain statistical significance. Thus in truly large numbers of observations, it is paradoxically easy to find significant correlations, in large numbers, which still do not lead to causal theories (see: spurious correlation), and which by their collective number, might lead to obfuscation as well.

The law can be rephrased as "large numbers also deceive", something which is counter-intuitive to a descriptive statistician. More concretely, skeptic Penn Jillette has said, "Million-to-one odds happen eight times a day in New York City " (population about 8,000,000).

Examples

Graphs of probability P of not observing independent events each of probability 1/n after n Bernoulli trials, and 1 − P  vs n. As n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 1/e.

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).

For a sample of only 1000 independent trials, however, the probability that the event does not happen in any of them, even once (improbability), is only 0.999¹⁰⁰⁰ ≈ 0.3677, or 36.77%. Then, the probability that the event does happen, at least once, in 1000 trials is ( 1 − 0.999¹⁰⁰⁰ ≈ 0.6323, or ) 63.23%. This means that this "unlikely event" has a probability of 63.23% of happening if 1000 independent trials are conducted. If the number of trials were increased to 10,000, the probability of it happening at least once in 10,000 trials rises to ( 1 − 0.999¹⁰⁰⁰⁰ ≈ 0.99995, or ) 99.995%. In other words, a highly unlikely event, given enough independent trials with some fixed number of draws per trial, is even more likely to occur.

For an event X that occurs with very low probability of 0.0000001% (in any single sample, see also almost never), considering 1,000,000,000 as a "truly large" number of independent samples gives the probability of occurrence of X equal to 1 − 0.999999999^1000000000 ≈ 0.63 = 63% and a number of independent samples equal to the size of the human population (in 2021) gives probability of event X: 1 − 0.999999999^7900000000 ≈ 0.9996 = 99.96%.

These calculations can be formalized in mathematical language as: "the probability of an unlikely event X happening in N independent trials can become arbitrarily near to 1, no matter how small the probability of the event X in one single trial is, provided that N is truly large."

For example, where the probability of unlikely event X is not a small constant but decreased in function of N, see graph.

In high availability systems even very unlikely events have to be taken into consideration, in series systems even when the probability of failure for single element is very low after connecting them in large numbers probability of whole system failure raises (to make system failures less probable redundancy can be used - in such parallel systems even highly unreliable redundant parts connected in large numbers raise the probability of not breaking to required high level).

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 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, the opposite may also be true when they think they need more analysis of their losses to achieve fine tuning of their playing system). 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 – "selective memory bias in either direction").

References

• Weisstein, Eric W.. "Law of truly large numbers". http://mathworld.wolfram.com/LawofTrulyLargeNumbers.html. 
• Diaconis, P.; Mosteller, F. (1989). "Methods of Studying Coincidences". Journal of the American Statistical Association 84 (408): 853–61. doi:10.2307/2290058. Archived from the original on 2010-07-12. https://web.archive.org/web/20100712091914/http://stat.stanford.edu/~cgates/PERSI/papers/mosteller89.pdf. Retrieved 2009-04-28. 
• Everitt, B.S. (2002). Cambridge Dictionary of Statistics (2nd ed.). ISBN 978-0521810999. 
• David J. Hand, (2014), The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day

External links

• Math Explains Likely Long Shots, Miracles and Winning the Lottery (Excerpt) in Scientific American by David Hand 2014
• skepdic.com on the Law of Truly Large Numbers
• on the Law of Truly Large Numbers
• The On-Line Encyclopedia of Integer Sequences – related integer sequence