1.5. Law of large numbers

The law of large numbers is one of the most important theorems in probability theory. It demonstrates and proves the fundamental relationship between the concepts of probability and frequency. In a way, it provides the bridge between probability theory and the real world.

Formal definition

As the number of identically distributed, randomly generated variables increases, their sample mean (average) approaches their theoretical mean. Another way to interpret it is as a probabilistic process is repeated a large number of times, the relative frequencies of its possible outcomes will get closer and closer to their respective probabilities.

How the law would manifest in real-world experiments?

1. Flipping a regular coin many times results in approximately 50% heads and 50% tails frequency since the probabilities of those outcomes are both 0.5.

2. Or throwing a fair dice will lead that each probability of each face to be drawn being 1/6.