Law of Large Numbers

The law of large numbers is a term used in probability theory to describe the result of performing the same experiment numerous times. According to the law of large numbers, the average value of the combined results should be close to the expected value, and the average value will become closer to the expected value as more trials are performed. For example, when a coin is flipped once, the expected value of it landing on heads or tails is one half. One could flip the coin five times and the coin could land on heads five times in a row, but the expected value remains one half. The more times the coin is flipped, the closer the expected value of landing on heads or tails will be to one half. In other words, although the coin may land on the same side several times in a row, as the number of flips approaches infinity, the proportion of heads after n flips will converge towards one half.

The origin of the law of large numbers stems from the 16th century Italian mathematician Gerolamo Cardano, who stated that the accuracy of empirical statistics tends to improve with the number of trials (1). The law of large numbers was first demonstrated by Jacob Bernoulli and was published in his Art of Conjecture in 1773.


(1) Mlodinow, Leonard (2008). “The Drunkard’s Walk: How Randomness Rules Our Lives“. New York: Pantheon Books.