The Long-Run Average

Probability · Axiom Academy

Discover expected value through games, gambling, and the law of large numbers. Look at these two dice games. Which one would you rather play? Click to reveal the expected values. Step 2: Play and Watch the Average Click the button to play Game A repeatedly. Watch how the running average changes! A game is "fair" if the expected value equals the cost to play. Let's explore! The long-run average of a random variable. Calculated as E[X] = Σ x · P(X = x), summing each value times its probability. If you repeat a random experiment many times, the average outcome converges to E[X]. It's not what you "expect" on one trial - it's the long-run average! • Determining if a game is fair • Making business decisions under uncertainty • Insurance pricing and risk assessment

This is the written version of the interactive lesson above. See the full Probability course.