When a random phenomenon is repeated many times, the proportion of trials on which an outcome occurs eventually approaches the probability of the outcome. If the outcomes are numerical, the average of the observed outcomes eventually approaches the expected value. Sometimes we express the random outcome as X, a random variable; then the expected value is also called the mean of X.

In the applet below select the number of "fair" dice to roll by clicking the More dice or Fewer dice buttons. An outcome for random variable X is the total of the dice for a given roll. Fill in the number of rolls and click the Roll dice button. The cumulative average is plotted after each roll.

Click the checkbox to show the expected value line. Notice how over the long-run the average of the dice totals becomes closer to the expected value.