The law of truly large numbers (a
statistical
Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industria ...
adage
An adage (; Latin: adagium) is a memorable and usually philosophical aphorism that communicates an important truth derived from experience, custom, or both, and that many people consider true and credible because of its longeval tradition, i.e. ...
), attributed to
Persi Diaconis
Persi Warren Diaconis (; born January 31, 1945) is an American mathematician of Greek descent and former professional magician. He is the Mary V. Sunseri Professor of Statistics and Mathematics at Stanford University.
He is particularly know ...
and
Frederick Mosteller
Charles Frederick Mosteller (December 24, 1916 – July 23, 2006) was an American mathematician, considered one of the most eminent statisticians of the 20th century. He was the founding chairman of Harvard's statistics department from 19 ...
, 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
Pseudoscience consists of statements, beliefs, or practices that claim to be both scientific and factual but are incompatible with the scientific method. Pseudoscience is often characterized by contradictory, exaggerated or unfalsifiable claim ...
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
Penn Fraser Jillette (born March 5, 1955) is an American magician, actor, musician, inventor, television presenter, and author, best known for his work with fellow magician Teller as half of the team Penn & Teller. The duo has been featured ...
has said, "Million-to-one odds happen eight times a day in
New York
New York most commonly refers to:
* New York City, the most populous city in the United States, located in the state of New York
* New York (state), a state in the northeastern United States
New York may also refer to:
Film and television
* '' ...
" (population about 8,000,000).
Examples
![Bernoulli trial sequence](https://upload.wikimedia.org/wikipedia/commons/d/d1/Bernoulli_trial_sequence.svg)
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 0.999
1000 ≈0.3677, or 36.77%. Then, the probability that the event does happen, at least once, in 1000 trials is 0.999
1000 ≈0.6323, 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 0.999
10000 ≈0.99995, 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
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
), considering 1,000,000,000 as a "truly large" number of independent samples gives the probability of occurrence of X equal to and a number of independent samples equal to the size of the
human population
Humans (''Homo sapiens'') are the most abundant and widespread species of primate, characterized by bipedalism and exceptional cognitive skills due to a large and complex brain. This has enabled the development of advanced tools, culture, ...
(in 2021) gives probability of event X:
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
sexual reproduction
Sexual reproduction is a type of reproduction that involves a complex life cycle in which a gamete ( haploid reproductive cells, such as a sperm or egg cell) with a single set of chromosomes combines with another gamete to produce a zygote tha ...
, the chances for a
microscopic
The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens (optics), lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded a ...
, single
spermatozoon
A spermatozoon (; also spelled spermatozoön; ; ) is a motile sperm cell, or moving form of the haploid cell that is the male gamete. A spermatozoon joins an ovum to form a zygote. (A zygote is a single cell, with a complete set of chromosomes, t ...
to reach the
ovum
The egg cell, or ovum (plural ova), is the female reproductive cell, or gamete, in most anisogamous organisms (organisms that reproduce sexually with a larger, female gamete and a smaller, male one). The term is used when the female gamete is ...
in order to
fertilize it is very small. Thus, in every encounter, spermatozoa are released in numbers of millions at once (in
mammal
Mammals () are a group of vertebrate animals constituting the class Mammalia (), characterized by the presence of mammary glands which in females produce milk for feeding (nursing) their young, a neocortex (a region of the brain), fur or ...
s), raising the opportunities of fecundation to a nearly-certain event.
In
high availability
High availability (HA) is a characteristic of a system which aims to ensure an agreed level of operational performance, usually uptime, for a higher than normal period.
Modernization has resulted in an increased reliance on these systems. Fo ...
systems even very unlikely events have to be taken into consideration (to make system failures less probable
redundancy can be used).
In criticism of pseudoscience
The law comes up in criticism of
pseudoscience
Pseudoscience consists of statements, beliefs, or practices that claim to be both scientific and factual but are incompatible with the scientific method. Pseudoscience is often characterized by contradictory, exaggerated or falsifiability, unfa ...
and is sometimes called the
Jeane Dixon effect (see also
Postdiction
Postdiction involves explanation after the fact.
In skepticism, it is considered an effect of hindsight bias that explains claimed predictions of significant events such as plane crashes and natural disasters. In religious contexts, theologian ...
). 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
Confirmation bias is the tendency to search for, interpret, favor, and recall information in a way that confirms or supports one's prior beliefs or values. People display this bias when they select information that supports their views, ignoring ...
). Humans can be susceptible to this fallacy.
Another similar manifestation of the law can be found in
gambling
Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three el ...
, where gamblers tend to remember their wins and forget their losses,
[Daniel Freeman, Jason Freeman, 2009, London, ''"Know Your Mind: Everyday Emotional and Psychological Problems and How to Overcome Them"'' p. 41] 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, 2002, Illinois, ''The Psychodynamics and Psychology of Gambling: The Gambler's Mind'' vol. I, p. 129]). 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").
]
See also
Notes
References
*
*
*{{cite book , last= Everitt , first= B.S. , year= 2002 , title= Cambridge Dictionary of Statistics , edition= 2nd , 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
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
by David Hand 2014
skepdic.com on the ''Law of Truly Large Numbers''
The On-Line Encyclopedia of Integer Sequences
– related integer sequence
Probability theory
Statistical laws