Expected Value Explained: Formula, Examples & Applications
Expected value is the theoretical long-run average of a random variable. It answers: "If I repeated this experiment thousands of times, what would the average outcome be?"
It is the cornerstone of rational decision-making under uncertainty — from insurance pricing to poker strategy to investment analysis.
Formula
E(X) = Σ xᵢ · P(xᵢ)
Building Intuition
Roll a fair d6. E(X) = 1×(1/6) + 2×(1/6) + 3×(1/6) + 4×(1/6) + 5×(1/6) + 6×(1/6) = 21/6 = 3.5. You never actually roll 3.5, but if you rolled thousands of times and averaged the results, you'd get closer and closer to 3.5.
For a fair coin: E(heads) = 1×0.5 + 0×0.5 = 0.5 — half a head on average. This is the long-run fraction of flips that come up heads.
Expected Value in Decision Making
A positive EV bet is worth taking (in the long run); a negative EV bet loses money on average. This is the mathematical basis for all rational gambling analysis.
Example game: pay $1 to roll a d6; win $5 if you roll a 6, $0 otherwise. E(gain) = 5×(1/6) + 0×(5/6) = 5/6 ≈ $0.83. E(profit) = $0.83 − $1.00 = −$0.17 per play. Negative EV — avoid this game.
Insurance example: a $500 policy with 2% annual claim probability and $30,000 average claim has EV for insurer = 500 − 0.02×30,000 = 500 − 600 = −$100. The insurer loses on average — unless the $500 premium correctly prices the risk. In practice, premiums are set with a profit margin.
Casino Games — Expected Value
All standard casino games have negative expected value for the player:
| Game | House Edge | EV per $100 bet |
|---|---|---|
| American Roulette (single number) | 5.26% | −$5.26 |
| European Roulette (single number) | 2.70% | −$2.70 |
| Blackjack (basic strategy) | 0.5% | −$0.50 |
| Craps (pass line) | 1.41% | −$1.41 |
| Slot machines (varies) | 5–15% | −$5 to −$15 |
| Video poker (optimal play) | 0.5–1% | −$0.50 to −$1.00 |
Expected Value in Finance
Expected return on an investment: E(R) = Σ rᵢ × P(scenario i). Example: an investment returns +20% in a bull market (probability 60%) and −10% in a bear market (40%). E(R) = 0.20×0.60 + (−0.10)×0.40 = 0.12 − 0.04 = 8% expected annual return.
Variance and standard deviation measure risk around the expected return. High EV + high variance = potentially good but volatile. Risk-averse investors accept lower EV for lower variance (diversification).
Linearity of Expectation
E(aX + bY) = a·E(X) + b·E(Y) — this holds for ANY random variables X and Y, even if they are dependent.
This is one of probability's most powerful tools. Example: expected sum of two d6 = E(d6) + E(d6) = 3.5 + 3.5 = 7, no need to enumerate all 36 outcomes.
Expected number of heads in 100 fair coin flips = 100 × E(one flip) = 100 × 0.5 = 50, by linearity.