  The strong law of large numbers states that where each trial has an expected value of \(\mu\), for any arbitrarily small real number \(\epsilon \gt 0\), as the number of trials increases, the probability that the gambler's average winnings per trial fall within \(\epsilon\) of \(\mu\) converges to 1. In other words, as the number of repetitions of a gamble approaches infinity, the average gain per trial will become arbitrarily close to the gamble's expected value with probability 1. So in the long run, the average value associated with a gamble is virtually certain to equal its expected value.