Expected Value

Pre-Algebra · Axiom Academy

One number that captures every outcome at once — the probability-weighted average of what can happen. 1. Weight Each Outcome by Its Chance Take the coin game: + 2 on heads, − 1 on tails, each with probability 0.5 . To get the expected value, multiply every outcome by its probability and add the pieces up. A 50% chance of 2 contributes 1 ; a 50% chance of losing 1 contributes − 0.50 . Each payoff, scaled by its probability, then summed 2. When Outcomes Aren't Equally Likely Most situations aren't 50/50. Consider a prize wheel with three results, and notice the probabilities still add to 1: The plain average of 0 , 10 , and 50 is about 20 — but that pretends all three are equally likely. The expected value weights each by its real chance, so the common 0 outcome dominates and pulls the answer down to 8 . There's a picture for this. Place a weight at each payoff, with heavier weights for the more likely outcomes. The expected value is the single point where the beam balances — the center of mass of the outcomes. A bigger probability is a heavier weight, so it tugs the balance point toward its payoff more strongly. The beam settles at 8 — closer to the likely 0 than to the rare 50 . 3. The Average You'd Actually See Expected value isn't just a formula — it's a prediction . Roll a fair die: each face 1 through 6 has probability , so the expected value is the weighted average of the faces.

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