How does expected value describe the long-run average of repeated trials?

In probability and statistics, the expected value—often denoted as $E(x)$ or $\mu$—serves as the weighted average of all possible outcomes of a random variable, where each outcome is weighted by its probability of occurring.12

While the expected value does not necessarily represent a value that can occur in a single trial, it describes the long-run average of repeated trials through the law of large numbers. This theorem establishes that as an experiment is repeated an increasing number of times under identical, independent conditions, the empirical average of the results will converge toward the expected value.3

Essentially, if you perform a statistical experiment many times, the average of those outcomes will get closer and closer to the expected value as the number of trials increases. This principle allows mathematicians and analysts to predict long-term outcomes, such as the profitability of investments or the structure of games in a casino, by calculating the weighted mean of all potential results.31