The law of large numbers is a major theorem in the mathematical study of probability. It can be applied to statistical analysis and business growth. This theorem states that when a certain experiment is executed enough number of times, the average result of those executions will move closer to the value we are expecting. What does this mean?
The Toss Of The Coin
One of the most common examples to help illustrate the practical application of this theorem is a simple game of a coin toss. Let's suppose you and your friend are playing a game of coin toss. If the coin lands on the head, you win, and if the coin lands on tails, your friend wins. If the coin is not rigged or a trick coin, there is a theoretical probability of 50% for the coin to land on either side. Let's say your friend lands tails every single time and beats you.
Does that mean that the next ten times, it's going to land on heads, and you will win? No, that is not the case. The law of large numbers says that continuing to play indefinitely, the number of heads and tails should eventually even out and get close to 50%, just like the theoretical probability suggested at the start of the game.
One of the common misconceptions is that with each new throw, there is going to be a probabilistic tendency for a coin to show the opposite side, but this is not true. Each and every coin toss is viewed as an individual and independent event with equal opportunity of landing on both sides. This common misconception is sometimes also called a gambler's fallacy.
There is simply no way for us to predict the outcome of a single coin flip. The law of large numbers is only telling us about the eventual outcome if we continue with the same observable action. It won't happen in ten or twenty tries, but after a few million attempts, you will be less likely to flip just one side for 80 or 90%. This same theorem can be applied to a variety of statistical surveys. We can never successfully perform a survey on the entirety of the population we are investigating. Still, we can rest assured that the more people we survey, the more accurate the representation is going to be at the end.
The History Of The Law Of Large Numbers
The Italian mathematician Gerolamo Cardano was the first one who observed The Law of Large Numbers, but he never tried to prove his observations. Jacob Bernoulli, a mathematician from Switzerland, was the one who offered proof and laid the foundation of the study of this theorem. He initially called it his "Golden Theorem".
Describing an endless game of chance with only two possible outcomes, Bernoulli proved that as the numbers of repetitions increased, the chances of those two outcomes would come closer and closer together, eventually evening out. He worked over 20 years in hopes of developing the mathematical proof needed for this theorem. The results of Bernoulli's work were published in his magnum opus "Ars Conjectandi, "in the year 1713.
About the Author
Antonia is a sociologist and an anglicist by education, but a writer and a behavior enthusiast by inclination. If she's not writing, editing or reading, you can usually find her snuggling with her huge dog or being obsessed with a new true-crime podcast. She also has a (questionably) healthy appreciation for avocados and Seinfeld.
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