The law of large numbers is a fundamental concept in probability theory. It states that the average of the results from a large sample size will tend to be closer to the expected value than the average of the results from a small sample size. This concept is sometimes referred to as the “Law of Averages.”

Essentially, it means that a sample size that is sufficiently large will better represent the characteristic of the entire population than a smaller sample size would. This is because the sample size inversely affects the amount of randomness present in the data. A larger sample size more accurately reflects the population as a whole, due to the fact that deviating samples are averaged out.

In the field of finance, the law of large numbers is used to understand the relationship between business scale and growth rate. This relationship states that as the scale of a business grows, the ability to attain percentage growth goals becomes increasingly difficult; because the underlying dollar amounts become larger and those goals become less realistic.

For example, if a company has a goal to increase profits by 10%, they may find that it is much easier to do when they have a smaller base of profits (e.g., 50000$) versus a larger one (e.g., 100000$). This is because achieving the same percentage increase for either number requires the achievement of a larger dollar increase for the larger base (10000 $ growth in profits for the former versus 20000 $ growth for the latter).

The law of large numbers applies to more than just economics and finance. It is used in many areas of life where large numbers and statistics are used, including public opinion polling, as well as in experiments done in natural sciences.

In short, the law of large numbers provides us with a fundamental concept in probability theory and is used to explain the relationship between scale and growth rates in business. The principle it espouses is that, with a large enough sample size, we can get a better handle on a population's characteristics, and that we are more likely to see results if a sufficient amount of data is collected.