"[…] probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; law of large numbers […]" (Jacob Bernoulli, "Ars Conjectandi" ["The Art of Conjecturing"], 1713)
"Things of all kinds are subject to a universal law which may be called the law of large numbers. It consists in the fact that, if one observes very considerable numbers of events of the same nature, dependent on constant causes and causes which vary irregularly, sometimes in one direction, sometimes in the other, it is to say without their variation being progressive in any definite direction, one shall find, between these numbers, relations which are almost constant." (Siméon-Denis Poisson, "Poisson’s Law of Large Numbers", 1837)
"It is a common fallacy to believe that the law of large numbers acts as a force endowed with memory seeking to return to the original state, and many wrong conclusions have been drawn from this assumption." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950)
“We know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?” (Albert Claude, [Nobel Prize Lecture], 1974)
"The law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen." (Frederick Mosteller, Methods for Studying Coincidences Journal of the American Statistical Association, Volume 84, 1989)
"All the law [of large numbers] tells us is that the average of a large number of throws will be more likely than the average of a small number of throws to differ from the true average by less than some stated amount. And there will always be a possibility that the observed result will differ from the true average by a larger amount than the specified bound."
"Jacob Bernoulli's theorem for calculating probabilities a posteriori is known as the Law of Large Numbers. Contrary to the popular view, this law does not provide a method for validating observed facts, which are only an incomplete representation of the whole truth. Nor does it say that an increasing number of observations will increase the probability that what you see is what you are going to get. The law is not a design for improving the quality of empirical tests […]." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"The Law of Large Numbers does not tell you that the average of your throws will approach 50% as you increase the number of throws; simple mathematics can tell you that, sparing you the tedious business of tossing the coin over and over. Rather, the law states that increasing the number of throws will correspondingly increase the probability that the ratio of heads thrown to total throws will vary from 50% by less than some stated amount, no matter how small. The word 'vary' is what matters. The search is not for the true mean of 50% but for the probability that the error between the observed average and the true average will be less than, say, 2% - in other words, that increasing the number of throws will increase the probability that the observed average will fall within 2% of the true average." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"The law of small numbers is not really a law. It is a sarcastic name describing the misguided attempt to apply the law of large numbers when the numbers aren't large." (Leonard Mlodinow, "The Drunkard’s Walk: How Randomness Rules Our Lives", 2008)
"The law of large numbers is a law of mathematical statistics. It states that when random samples are sufficiently large they match the population extremely closely. […] The 'law' of small numbers is a widespread human misconception that even small samples match the population closely." (Geoff Cumming, "Understanding the New Statistics", 2012)
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