Law of large numbers

The **law of large numbers**, or LLN for short, is a theorem from statistics. It states that if a random process is repeatedly observed, then the average of the observed values will be stable in the long run. This means that as the number of observations increases, the average of the observed values will get closer and closer to the expected value.

For example, when rolling dice, the numbers 1, 2, 3, 4, 5 and 6 are possible outcomes. They are all equally likely.  The population mean (or "expected value") of the outcomes is:

- (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5.

The following graph shows the results of an experiment of rolls of a die. In this experiment, it can be seen that the average of die rolls varies wildly at first, but as predicted by the LLN, the average stabilizes around the expected value of 3.5 as the number of observations become large.

## History

Jacob Bernoulli first described the LLN. He said the concept was so simple that even the stupidest man instinctively knows it is true. Despite this, it took him over 20 years to develop a good mathematical proof. Once he had found it, he published the proof in _Ars Conjectandi_ (The Art of Conjecturing) in 1713.  He named this his "golden theorem". It became generally known as "Bernoulli's theorem" (not to be confused with the law in physics with the same name). In 1835, S.D. Poisson further described it under the name "la loi des grands nombres" (the law of large numbers).  Thereafter, it was known under both names, but the "law of large numbers" is most frequently used.

Law of large numbers of Googol to a Maximusmultillion (limit)
 Googol: 10^100
 Googolchime: 10^1,000
 Googoltoll: 10^10,000
 Googolgong: 10^100,000
 Maximusmillion: 10^1,000,000
 Goospolplex: 10^10,000,000
 Googolbong: 10^100,000,000
 Maximusbillion: 10^1,000,000,000
 Trialogue: 10^10,000,000,000
 Googolthrong: 10^100,000,000,000
 Maximustrillion: 10^1,000,000,000,000
 Trdoocolplex: 10^10,000,000,000,000
 Googolgandingan: 10^100,000,000,000,000
 Maximusquadrillion: 10^1,000,000,000,000,000
 Maximusbyllion: 10^10,000,000,000,000,000
 Googolquintigong: 10^100,000,000,000,000,000
 Maximusquintillion: 10^1,000,000,000,000,000,000
 Guppyplexichunk: 10^10,000,000,000,000,000,000
 Guppyplex: 10^100,000,000,000,000,000,000
 Maximussextillion: 10^1,000,000,000,000,000,000,000
 Maximusseptillion: 10^10^24
 Minnowplex: 10^10^25
 Maximusoctillion: 10^10^27
 Maximusnonillion: 10^10^30
 Maximustryllion: 10^10^32
 Maximusdecillion: 10^10^33
 Gobyplex: 10^10^35
 Maximusundecillion: 10^10^36
 Maximusduodecillion: 10^10^39
 Maximustredecillion: 10^10^42
 Maximusquattuordecillion: 10^10^45
 Maximusquindecillion: 10^10^48
 Gogolplex: 10^10^50
 Maximussexdecillion: 10^10^51
 Maximusseptendecillion: 10^10^54
 Maximusoctodecillion: 10^10^57
 Maximusnovemdecillion: 10^10^60
 Maximusvigintillion: 10^10^63
 Prawn-plex: 10^10^65
 Maximusunvigintillion: 10^10^66
 Maximusduovigintillion: 10^10^69
 Maximustrevigintillion: 10^10^72
 Lightweight-plex / Maximusquattuorvigintillion: 10^10^75
 Maximusquinvigintillion: 10^10^78
 Maximussexvigintillion: 10^10^81
 Maximusseptenvigintillion: 10^10^84
 Twerpuloid-plex: 10^10^85
 Maximusoctovigintillion: 10^10^87
 Maximusnovemvigintillion: 10^10^90
 Maximustrigintillion: 10^10^93
 Googolplex: 10^10^100
 Googoltyplex: 10^10^101
 Eleventyplex: 10^10^110
 Maximusquadragintillion: 10^10^123
 Maximusquinquagintillion: 10^10^153
 Maximussexagintillion: 10^10^183
 Gargoogol-plexed: 10^10^200
 Maximusseptuagintillion: 10^10^213
 Maximusoctogintillion: 10^10^243
 Maximusnonagintillion: 10^10^273
 Ecetonplex / Maximuscentillion: 10^10^303
 Googolplexiding: 10^10^500
 Maximusducentillion: 10^10^603
 Maximustrecentillion: 10^10^903
 Googolplexichime: 10^10^1,000
 Maximusquadringentillion: 10^10^1,203
 Maximusquingentillion: 10^10^1,503
 Maximussescentillion: 10^10^1,803
 Maximusseptingentillion: 10^10^2,103
 Maximusoctingentillion: 10^10^2,403
 Maximusnongentillion: 10^10^2,703
 Maximusmillillion: 10^10^3,003
 Googolplexibell: 10^10^5,000
 Maximusdumillillion: 10^10^6,003
 Maximustrimillillion: 10^10^9,003
 Googolplexitoll: 10^10^10,000
 Maximusquadrimillillion: 10^10^12,003
 Maximusquinmillillion: 10^10^15,003
 Maximussexmillillion: 10^10^18,003
 Maximusseptimillillion: 10^10^21,003
 Maximusoctimillillion: 10^10^24,003
 Maximusnonimillillion: 10^10^27,003
 Maximusmyrillion / Maximusdecimillillion: 10^10^30,003
 Googolplexigong: 10^10^100,000
 Maximuscentimillillion: 10^10^300,003
 Milliduplexion: 10^10^1,000,000
 Maximusmicrillion: 10^10^3,000,003
 Googolplexibong: 10^10^100,000,000
 Maximusnanillion: 10^10^3,000,000,003
 Googolplexithrong: 10^10^100,000,000,000
 Maximuspicillion: 10^10^3,000,000,000,003
 Maximusfemtillion: 10^10^(3*10^15+3)
 Maximusattillion: 10^10^(3*10^18+3)
 Maximuszeptillion: 10^10^(3*10^21+3)
 Maximusyoctillion: 10^10^(3*10^24+3)
 Maximusxonillion: 10^10^(3*10^27+3)
 Maximusvecillion: 10^10^(3*10^30+3)
 Maximusmecillion: 10^10^(3*10^33+3)
 Maximusduecillion: 10^10^(3*10^36+3)
 Maximustrecillion: 10^10^(3*10^39+3)
 Maximustetrecillion: 10^10^(3*10^42+3)
 Maximuspentecillion: 10^10^(3*10^45+3)
 Maximushexecillion: 10^10^(3*10^48+3)
 Maximusheptecillion: 10^10^(3*10^51+3)
 Maximusoctecillion: 10^10^(3*10^54+3)
 Maximusennecillion: 10^10^(3*10^57+3)
 Maximusicosillion: 10^10^(3*10^60+3)
 Maximusmeicosillion: 10^10^(3*10^63+3)
 Maximusdueicosillion: 10^10^(3*10^66+3)
 Maximustrioicosillion: 10^10^(3*10^69+3)
 Maximustetreicosillion: 10^10^(3*10^72+3)
 Maximuspenteicosillion: 10^10^(3*10^75+3)
 Maximushexeicosillion: 10^10^(3*10^78+3)
 Maximushepteicosillion: 10^10^(3*10^81+3)
 Maximusocteicosillion: 10^10^(3*10^84+3)
 Maximusenneicosillion: 10^10^(3*10^87+3)
 Maximustriacontillion: 10^10^(3*10^90+3)
 Maximustetracontillion: 10^10^(3*10^120+3)
 Maximuspentacontillion: 10^10^(3*10^150+3)
 Maximushexacontillion: 10^10^(3*10^180+3)
 Maximusheptacontillion: 10^10^(3*10^210+3)
 Maximusoctacontillion: 10^10^(3*10^240+3)
 Maximusennacontillion: 10^10^(3*10^270+3)
 Maximushectillion: 10^10^(3*10^300+3)
 Maximusdohectillion: 10^10^(3*10^600+3)
 Maximustriahectillion: 10^10^(3*10^900+3)
 Maximustetrahectillion: 10^10^(3*10^1,200+3)
 Maximuspentahectillion: 10^10^(3*10^1,500+3)
 Maximushexahectillion: 10^10^(3*10^1,800+3)
 Maximusheptahectillion: 10^10^(3*10^2,100+3)
 Maximusoctahectillion: 10^10^(3*10^2,400+3)
 Maximusennahectillion: 10^10^(3*10^2,700+3)
 Maximuskillillion: 10^10^(3*10^3,000+3)
 Maximusvecekillillion: 10^10^(3*10^30,000+3)
 Maximushectekillillion: 10^10^(3*10^300,000+3)
 Maximusmegillion: 10^10^(3*10^3,000,000+3)
 Maximusvecemegillion: 10^10^(3*10^30,000,000+3)
 Maximushectemegillion: 10^10^(3*10^300,000,000+3)
 Maximusgigillion: 10^10^(3*10^3,000,000,000+3)
 Maximusterillion: 10^10^(3*10^3,000,000,000,000+3)
 Maximuspetillion: 10^10^(3*10^(3*10^15)+3)
 Maximusexillion: 10^10^(3*10^(3*10^18)+3)
 Maximuszettillion: 10^10^(3*10^(3*10^21)+3)
 Maximusyottillion: 10^10^(3*10^(3*10^24)+3)
 Maximusxennillion: 10^10^(3*10^(3*10^27)+3)
 Maximusdakillion: 10^10^(3*10^(3*10^30)+3)
 Maximushendillion: 10^10^(3*10^(3*10^33)+3)
 Maximusdokillion: 10^10^(3*10^(3*10^36)+3)
 Maximustradakillion: 10^10^(3*10^(3*10^39)+3)
 Maximustedakillion: 10^10^(3*10^(3*10^42)+3)
 Maximuspedakillion: 10^10^(3*10^(3*10^45)+3)
 Maximusexdakillion: 10^10^(3*10^(3*10^48)+3)
 Maximuszedakillion: 10^10^(3*10^(3*10^51)+3)
 Maximusyodakillion: 10^10^(3*10^(3*10^54)+3)
 Maximusnedakillion: 10^10^(3*10^(3*10^57)+3)
 Maximusikillion: 10^10^(3*10^(3*10^60)+3)
 Maximusikenillion: 10^10^(3*10^(3*10^63)+3)
 Maximusicodillion: 10^10^(3*10^(3*10^66)+3)
 Maximusictrillion: 10^10^(3*10^(3*10^69)+3)
 Maximusicterillion: 10^10^(3*10^(3*10^72)+3)
 Maximusicpetillion: 10^10^(3*10^(3*10^75)+3)
 Maximusikectillion: 10^10^(3*10^(3*10^78)+3)
 Maximusiczetillion: 10^10^(3*10^(3*10^81)+3)
 Maximusikyotillion: 10^10^(3*10^(3*10^84)+3)
 Maximusicxenillion: 10^10^(3*10^(3*10^87)+3)
 Maximustrakillion: 10^10^(3*10^(3*10^90)+3)
 Maximustekillion: 10^10^(3*10^(3*10^120)+3)
 Maximuspekillion: 10^10^(3*10^(3*10^150)+3)
 Maximusexakillion: 10^10^(3*10^(3*10^180)+3)
 Maximuszakillion: 10^10^(3*10^(3*10^210)+3)
 Maximusyokillion: 10^10^(3*10^(3*10^240)+3)
 Maximusnekillion: 10^10^(3*10^(3*10^270)+3)
 Maximushotillion: 10^10^(3*10^(3*10^300)+3)
 Maximusbotillion: 10^10^(3*10^(3*10^600)+3)
 Maximustrotillion: 10^10^(3*10^(3*10^900)+3)
 Maximustotillion: 10^10^(3*10^(3*10^1,200)+3)
 Maximuspotillion: 10^10^(3*10^(3*10^1,500)+3)
 Maximusexotillion: 10^10^(3*10^(3*10^1,800)+3)
 Maximuszotillion: 10^10^(3*10^(3*10^2,100)+3)
 Maximusyootillion: 10^10^(3*10^(3*10^2,400)+3)
 Maximusnotillion: 10^10^(3*10^(3*10^2,700)+3)
 Maximuskalillion: 10^10^(3*10^(3*10^3,000)+3)
 Maximusdakalillion: 10^10^(3*10^(3*10^30,000)+3)
 Maximushotalillion: 10^10^(3*10^(3*10^300,000)+3)
 Maximusmejillion: 10^10^(3*10^(3*10^3,000,000)+3)
 Maximusgijillion: 10^10^(3*10^(3*10^3,000,000,000)+3)
 Maximusastillion: 10^10^(3*10^(3*10^(3*10^12))+3)
 Maximuslunillion: 10^10^(3*10^(3*10^(3*10^15))+3)
 Maximusfermillion: 10^10^(3*10^(3*10^(3*10^18))+3)
 Maximusjovillion: 10^10^(3*10^(3*10^(3*10^21))+3)
 Maximussolillion: 10^10^(3*10^(3*10^(3*10^24))+3)
 Maximusbetillion: 10^10^(3*10^(3*10^(3*10^27))+3)
 Maximusglocillion: 10^10^(3*10^(3*10^(3*10^30))+3)
 Maximusgaxillion: 10^10^(3*10^(3*10^(3*10^33))+3)
 Maximussupillion: 10^10^(3*10^(3*10^(3*10^36))+3)
 Maximusversillion: 10^10^(3*10^(3*10^(3*10^39))+3)
 Maximusmultillion: 10^10^(3*10^(3*10^(3*10^42))+3) (limit)
 etc.

Other mathematicians also contributed to make the law better. Some of them were Chebyshev (who proved a more general version of the law for averages), Markov, Borel, Cantelli and Kolmogorov.  After these studies, there are now two different forms of the law: One is called the "weak" law, and the other the "strong" law.  These forms do not describe different laws. They have different ways to describe the convergence of the observed or measured probability to the actual probability. The strong form of the law implies the weak one.

## Related pages

- Central limit theorem