How does the effective law of large numbers function optimally?

The law of large numbers is a fundamental principle in insurance and statistics, showing how predictions become more accurate with larger sample sizes. It functions optimally when exposures are independent and identically distributed. This means that the risks associated with individual exposures do not influence each other and follow the same probability distribution. As the number of exposures increases under these conditions, the average outcome (for example, the expected losses) becomes more stable, allowing insurers to more accurately predict overall losses and set appropriate premiums. This predictability is crucial for insurers to manage risk effectively and ensures that they can maintain solvency over time while covering claims.

Other factors, such as combining high-risk members or limiting exposures, can distort risk assessments or lead to higher variability in outcomes, undermining the effectiveness of the law of large numbers. Predicting future losses accurately is a result of applying this law correctly, but it does not define how the law itself operates optimally. Thus, the optimal functioning of the law of large numbers relies on independence and identical distribution of exposures to ensure that risk can be pooled effectively.