líkindalögmálið

Líkinda lögmálið, also known as the law of large numbers, is a fundamental theorem in probability theory and statistics. It describes the long-term stability of the average of a random variable. In simple terms, it states that as the number of trials of an experiment increases, the average of the results obtained will get closer and closer to the expected value.

The law essentially bridges the gap between theoretical probability and empirical observation. For example, if you flip a fair coin, the theoretical probability of getting heads is 0.5. However, in a small number of flips, you might observe a proportion of heads that is significantly different from 0.5. The law of large numbers asserts that as you perform more and more flips, the observed proportion of heads will converge towards 0.5.

There are two main forms of the law: the weak law of large numbers and the strong law of large numbers. The weak law states that the probability of the sample average deviating from the expected value by more than any small amount approaches zero as the sample size increases. The strong law is a more powerful statement, asserting that the sample average will converge to the expected value with probability one.

This principle has wide-ranging applications in various fields, including finance, insurance, gambling, and scientific research. It underpins the reliability of statistical inference and allows us to make predictions about the behavior of systems based on empirical data. It is crucial for understanding how random processes behave over time and for making sound decisions in the face of uncertainty.