Editing Law of large numbers (section)

{{Distinguish|Law of truly large numbers}}

{{Short description|Averages of repeated trials converge to the expected value}}

{{Probability fundamentals}}

[[File:Lawoflargenumbers.svg|An [[illustration]] of the law of large numbers using a particular run of rolls of a single [[Dice|die]].  As the number of rolls in this run increases, the average of the values of all the results approaches 3.5.  Although each run would show a distinctive shape over a small number of throws (at the left), over a large number of rolls (to the right) the shapes would be extremely similar.|thumb|right|286x286px]]

In [[probability theory]], the '''law of large numbers''' is a [[Law (mathematics)|mathematical law]] that states that the [[average]] of the results obtained from a large number of independent random samples converges to the true value, if it exists.<ref name=":0">{{Cite book|title=A Modern Introduction to Probability and Statistics| url=https://archive.org/details/modernintroducti00fmde|url-access=limited| last=Dekking|first=Michel| publisher=Springer| year=2005|isbn=9781852338961|pages=[https://archive.org/details/modernintroducti00fmde/page/n191 181]–190}}</ref> More formally, the law of large numbers states that given a sample of independent and identically distributed values, the [[Sample mean and covariance|sample mean]] converges to the true [[mean]]. 

The law of large numbers is important because it guarantees stable long-term results for the averages of some [[Randomness|random]] [[Event (probability theory)|events]].<ref name=":0" /><ref>{{Cite journal|last1=Yao|first1=Kai|last2=Gao|first2=Jinwu|date=2016|title=Law of Large Numbers for Uncertain Random Variables|journal=IEEE Transactions on Fuzzy Systems| volume=24| issue=3| pages=615–621| doi=10.1109/TFUZZ.2015.2466080| s2cid=2238905|issn=1063-6706}}</ref> For example, while a [[casino]] may lose [[money]] in a single spin of the [[roulette]] wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a ''large number'' of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the [[gambler's fallacy]]).

The law of large numbers only applies to the ''average'' of the results obtained from repeated trials and claims that this average converges to the expected value; it does not claim that the ''sum'' of ''n'' results gets close to the expected value times ''n'' as ''n'' increases.

Throughout its history, many mathematicians have refined this law. Today, the law of large numbers is used in many fields including statistics, probability theory, economics, and insurance.<ref name=":1">{{Cite web |last=Sedor |first=Kelly |title=The Law of Large Numbers and its Applications |url=https://www.lakeheadu.ca/sites/default/files/uploads/77/images/Sedor%20Kelly.pdf}}</ref>