Law of large numbers

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In probability theory, the law of large numbers (LLN) is a mathematical theorem that states that the average of the results obtained from a large number of independent and identical random samples converges to the true value, if it exists.^[1] More formally, the LLN states that given a sample of independent and identically distributed values, the sample mean converges to the true mean.

The LLN is important because it guarantees stable long-term results for the averages of some random events.^[1][2] 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 LLN 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 LLN is used in many fields including statistics, probability theory, economics, and insurance.^[3]