Law of Large Numbers
The Law of Large Numbers is a theorem in Probability Theory, which provides the output when the same experiment is conducted a large number of times.
According to the Law, when a particular experiment is conducted a large number of times, the average output of the results tends to move towards an expected value, and will continue to converge towards it as the experiment is conducted more number of times.
Explanation for the Layman
The Law of Large Numbers is one of the most intuitive laws in mathematics, but also often misunderstood.
Let us consider this example – there is a box full of 10 coins. Now it is known that the probability of a heads turning up on a coin is .5, or there’s a 50% chance that if a random coin is blindly pulled out of the box, it’s going to turn up as heads (and the same for tails). So consider that a box full of 10 coins is given a shake and one by one the coins are taken out and results recorded. The Law of Large Numbers says that greater the number of times the box is shaken and results recorded, the closer the average of the number of heads is going to be to 0.5 or 50% (the expected value).
Application of Law of Large Numbers
The Law of Large Numbers can be applied to the following fields:
- Statistical Surveys – During statistical surveys, although the entire population cannot be surveyed, it can be safely said that the greater the number of people who are surveyed, the closer the outcome will be to the actual mean
- Business – The Law of Large Numbers in Business is generally identified in relation to the percentage growth rates of Blue Chip companies where it is said that as the company becomes bigger, the same high growth rates become increasingly difficult to maintain
The Law of Large Numbers was first proved by the Swiss mathematician Jacob Bernoulli in his work published posthumously in 1713. He also called it “Law of Averages”, and explained it in detail with lots of examples. He highlighted the importance of conducting not one or two trials, but a large number of trials to help the “man on the street” make a judgement using this “law of averages”.
He stressed that sometimes the most ignorant man is aware, with the help of no previous instruction, that the more the number of observations are taken, the less the danger of him straying from the mark.
A more general version of the Law of Large Numbers for averages was proved after more than 100 years by Pafnuty Chebyshev, a Russian mathematician.
Carrying forward the example stated above of 10 coins in a box, it may be wrongly assumed that if I get Heads on the first three coins blindly taken out of the box after a random shake, there is a greater probability of a Tails turning up in the next draw to make up for the earlier Heads.
Although this may appeal to our intuitive sense, this is not true as the probability of a heads being turned up will always be 0.5 or 50% for each trial in a truly random trial.
This fallacy is commonly found amongst gamblers where a gambler may feel that because of a particular result has taken place more frequently thus far, it will take place less frequently in the future, or vice versa.