Law of the big numbers

The law of large numbers is a fundamental theorem of probability theory that indicates that if we repeat the same experiment many times (tending to infinity), the frequency of a certain event happening tends to be a constant.

Law of the big numbers

That is, the law of large numbers indicates that if the same test is carried out repeatedly (for example, tossing a coin, throwing a roulette wheel, etc.), the frequency with which a certain event will be repeated (that comes up heads or seal, the number 3 comes out black, etc) will approach a constant. This in turn will be the probability of this event occurring.

Origin of the law of large numbers

The law of large numbers was mentioned for the first time by the mathematician Gerolamo Cardamo, although without having any rigorous proof. Later, Jacob Bernoulli managed to make a complete demonstration in his work "Ars Conjectandi" in 1713. In the 1830s the mathematician Siméon Denis Poisson described in detail the law of large numbers, which came to perfect the theory. Other authors would also make later contributions.

Example of the law of large numbers

Suppose the following experiment: roll a common die. Now let’s consider the event that we get the number 1. As we know, the probability that the number 1 will come up is 1/6 (the die has 6 faces, one of them is one).

What does the law of large numbers tell us? It tells us that as we increase the number of repetitions of our experiment (we make more throws of the die), the frequency with which the event will be repeated (we get 1) will get closer to a constant, which will have an equal value to its probability (1/6 or 16.66%).

Possibly, in the first 10 or 20 launches, the frequency with which we get 1 will not be 16%, but another percentage such as 5% or 30%. But as we do more and more pitches (say 10,000), the frequency that the 1 appears will be very close to 16.66%.

In the following graphic we see an example of a real experiment where a die is rolled repeatedly. Here we can see how the relative frequency of drawing a certain number is changing.

As indicated by the law of large numbers, in the first launches the frequency is unstable, but as we increase the number of launches the frequency tends to stabilize at a certain number, which is the probability of the event occurring (in this case numbers from 1 to 6 since it is the throwing of a dice).

Misinterpretation of the law of large numbers

Many people misinterpret the law of large numbers believing that one event will tend to outweigh another. Thus, for example, they believe that since the probability that the number 1 will roll on the roll of a die should be close to 1/6, when the number 1 does not appear on the first 2 or 5 rolls, it is very likely that it will appear In the next. This is not true, since the law of large numbers only applies for many repetitions, so we can spend all day rolling a die and not reach the 1/6 frequency.

The roll of a die is an independent event and, therefore, when a certain number appears, this result does not affect the next roll. Only after thousands of repetitions will we be able to verify that the law of large numbers exists and that the relative frequency of getting a number (in our example 1) will be 1/6.

The misinterpretation of the theory can lead people (especially gamblers) to lose money and time.