The law of large numbers is a law that describes the probabilistic properties of an experiment when the number of trials is large. But note that the law of large numbers is not an empirical law, but in some additional conditions on the rigorously proved theorem, it is a law of nature and is usually not called a theorem but the law of large numbers "law". What we mean by a law of large numbers is usually a theorem of large numbers proved by a mathematician and named after the mathematician, such as Bernoulli's theorem of large numbers. The laws of large numbers are categorized into weak and strong laws of large numbers.

For example, in a randomized trial of repeated tosses of a coin, the number of times the coin comes up heads in n tosses is observed. The frequency of heads (the ratio of the number of heads to n) may be different for different n trials, but as the number of trials, n, gets larger, the frequency of heads will be roughly and gradually approaching 1/2. Another example is the weighing of a certain object, if there is no systematic bias in the scale, due to the accuracy of the scale and other factors, the same object is repeated many times, and many different weights may be obtained. values, but their arithmetic mean will generally be closer to the true weight of the object as the number of weighings increases.

Convergence almost everywhere is different from convergence by probability. Life example: the class begins and slowly everyone quiets down, this is convergence almost everywhere. The vast majority of students are quiet, but each one is not quiet at different times, which is convergence by probability.