What is it the Law of Large Numbers?
In probability theory and statistics, the law of large numbers is a theorem that explains the outcome of performing an experiment a large number of times. It provides a theoretical justification for the averaging process performed on experiments to obtain precise measurements. It states that if you repeat an experiment independently a large number of times and average the result, then the result should be close to the expected value.
Another way to say this is as the number of identically distributed (i. i. d.), randomly generated variables increases, their sample mean approaches the expected, or theoretical mean.
Examples
A researcher may take the average weights of many people to obtain an estimate of the average weight of humans. The average of many independently selected weights will be close to the true average weight with high probability.
In another example, when we flip a fair coin, the theoretical probability that the outcome will be heads is equal to 1/2. Therefore, according to the law of large numbers, the ratio of heads to tails in a large number of coin flips should be around 1/2. In other words, the ratio of heads after n flips will converge to 1/2 as n approaches infinity.
A Little Bit of History on the Law of Large Numbers…
Jakob Bernoulli first proved the law of large numbers in 1713. The Swiss mathematician and his colleagues were developing formal probability theory using games of chance. Bernoulli anticipated that a game of pure chance had only two outcomes, a success or a failure (e.g. Heads or Tails, 0 or 1, etc.).
Letting p represent the probability of a win, he considered the fraction of times a game of chance would be won in a large number of repetitions. It was believed that this fraction would eventually approach p. By showing that, as the number of repetitions increases, the probability of this fraction being within any distance from p approaches 1.
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References
- Wackerly, D.D., Mendenhall, W. and Scheaffer, R.L. (2008) Mathematical Statistics with Applications. 7th Edition, Thomson Learning, Inc., USA