The Law of Large Numbers and the Central Limit Theorem are understood in layman's terms as follows:

Sometimes statistical probability works like magic and is able to draw incredibly powerful conclusions from small amounts of data. We can go about predicting the number of votes in the U.S. presidential election simply by conducting a telephone survey of 1,000 Americans. By testing 100 pieces of chicken produced by a processing plant that supplies KFC with chicken for viruses (salmonella), we can conclude whether all meat products from that plant are safe.

The secret weapon behind this is the 2nd great protector of statistical probability: the central limit theorem. The 1st major guardian I have talked about in Lecture 3, "Investing for Money and Probability" is: the Law of Large Numbers.

The Central Limit Theorem is the "powerhouse" of many statistical activities, which share a common feature of using a sample to estimate the population, such as the classic example of public opinion polls that we often see.

In a randomized experiment, the result may be different each time it occurs, but if a large number of experiments are done (the number of experiments n->oo), the mean of the results of these experiments is close to the expected value (the overall mean). For example, if you flip a coin, you will find that heads or tails go up nearly half the time if you flip it enough times. Corresponding to sampling, if the sample size n tends to infinity, the mean of this sample will tend to the overall mean.

The Nature of the Law of Large Numbers

The first limit theorem in the history of probability theory belonged to Bernoulli, and was later called the Law of Large Numbers. In probability theory, it discusses the law that the arithmetic mean of a sequence of random variables converges to the arithmetic mean of the mathematical expectation of each of the random variables. In a large number of recurrences of random events, there is often an almost inevitable pattern, and this law is the law of large numbers.

In layman's terms, the theorem is that the frequency of a random event approximates its probability when the trial is repeated many times under conditions of constant trial. There is a certain inevitability contained in chance. We know that the law of large numbers studies the statistical regularity of random phenomena of a class of theorems, when we repeat a large number of times a certain same experiment, its final experimental results may be stabilized around a certain value.