The Law of Large Numbers and Central Limit Theorem

The law of large numbers states that in the long run, the ratio of outcomes will converge to the expected probabilities. This was initially proved for fair coin flips around 300 years ago and later extended to other systems. The central limit theorem suggests that when an event is repeated a large number of times, the outcomes will follow a bell curve, with deviations within a certain range. This phenomenon, based on the square root of the system size, is known as a Gaussian distribution and appears frequently in mathematics and science. Large deviation theory, on the other hand, deals with situations where average behavior is not of interest, and has its roots in the field of insurance.