 law of large numbers

Math.the theorem in probability theory that the number of successes increases as the number of experiments increases and approximates the probability times the number of experiments for a large number of experiments.[193540]
* * *
in statistics, the theorem that, as the number of identically distributed, randomly generated variables increases, their sample mean (average) approaches their theoretical mean.The law of large numbers was first proved by the Swiss mathematician Jakob Bernoulli (Bernoulli, Jakob) in 1713. He and his contemporaries were developing a formal probability theory with a view toward analyzing games of chance. Bernoulli envisaged an endless sequence of repetitions of a game of pure chance with only two outcomes, a win or a loss. Labeling the probability of a win p, Bernoulli considered the fraction of times that such a game would be won in a large number of repetitions. It was commonly believed that this fraction should eventually be close to p. This is what Bernoulli proved in a precise manner by showing that, as the number of repetitions increases indefinitely, the probability of this fraction being within any prespecified distance from p approaches 1.There is also a more general version of the law of large numbers for averages, proved more than a century later by the Russian mathematician Pafnuty Chebyshev (Chebyshev, Pafnuty Lvovich).The law of large numbers is closely related to what is commonly called the law of averages. In coin tossing, the law of large numbers stipulates that the fraction of heads will eventually be close to ^{1}/_{2}. Hence, if the first 10 tosses produce only 3 heads, it seems that some mystical force must somehow increase the probability of a head, producing a return of the fraction of heads to its ultimate limit of ^{1}/_{2}. Yet the law of large numbers requires no such mystical force. Indeed, the fraction of heads can take a very long time to approach ^{1}/_{2}(see figure—>). For example, to obtain a 95 percent probability that the fraction of heads falls between 0.47 and 0.53, the number of tosses must exceed 1,000. In other words, after 1,000 tosses, an initial shortfall of only 3 heads out of 10 tosses is swamped by results of the remaining 990 tosses.Richard Routledge* * *
Universalium. 2010.
Look at other dictionaries:
Law of large numbers — The law of large numbers (LLN) is a theorem in probability that describes the long term stability of the mean of a random variable. Given a random variable with a finite expected value, if its values are repeatedly sampled, as the number of these … Wikipedia
law of large numbers — Date: 1911 a theorem in mathematical statistics: the probability that the absolute value of the difference between the mean of a population sample and the mean of the population from which it is drawn is greater than an arbitrarily small amount… … New Collegiate Dictionary
Large numbers — This article is about large numbers in the sense of numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions. The term typically refers to large positive… … Wikipedia
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. It seeks to debunk one element of supposed supernatural… … Wikipedia
Names of large numbers — This article lists and discusses the usage and derivation of names of large numbers, together with their possible extensions. The following table lists those names of large numbers which are found in many English dictionaries and thus have a… … Wikipedia
Dirac large numbers hypothesis — Paul Dirac The Dirac large numbers hypothesis (LNH) is an observation made by Paul Dirac in 1937 relating ratios of size scales in the Universe to that of force scales. The ratios constitute very large, dimensionless numbers: some 40 orders of… … Wikipedia
Law of averages — The law of averages is a lay term used to express a belief that outcomes of a random event shall even out within a small sample.As invoked in everyday life, the law usually reflects bad statistics or wishful thinking rather than any mathematical… … Wikipedia
Large deviations theory — In Probability Theory, the Large Deviations Theory concerns the asymptotic behaviour of remote tails of sequences of probability distributions. Some basic ideas of the theory can be tracked back to Laplace and Cramér, although a clear unified… … Wikipedia
Law of Japan — Contents 1 Historical Developments 2 Sources of law 3 Precedent 4 Civil law 4.1 Contracts … Wikipedia
Law enforcement in the United Kingdom — Crime in the UK · Terrorism in the UK Topics … Wikipedia
Law enforcement jargon — refers to a large body of acronyms, abbreviations, codes and slang used by law enforcement personnel to provide quick concise descriptions of people, places, property and situations, in both spoken and written communication. These vary between… … Wikipedia