What expected value means
The expected value is the long-run average result of a random situation repeated many times. It is the basis for judging bets, insurance, investments and business decisions: a choice with a positive expected value makes money on average, one with a negative expected value loses it.
The formulas
- Expected value E(X) = Σ xᵢ × pᵢ
- Variance = Σ pᵢ × (xᵢ − E(X))²
- Standard deviation = √variance
Probabilities must add up to 100% when all outcomes are listed.
Worked example
A game where you:
| Outcome | Value | Probability |
|---|---|---|
| Big win | +$100 | 10% |
| Small win | +$20 | 20% |
| Loss | −$10 | 70% |
- Expected value: 100 × 0.10 + 20 × 0.20 − 10 × 0.70 = 10 + 4 − 7 = $7 per play.
- Variance: 0.10 × 93² + 0.20 × 13² + 0.70 × 17² = 864.9 + 33.8 + 202.3 = 1,101.
- Standard deviation: √1,101 ≈ $33.18.
On average you gain $7 per play, but individual results swing widely around that average.
Real-world examples
- Roulette: a $1 bet on red in American roulette wins $1 with probability 18/38 and loses $1 with probability 20/38. Expected value: −$0.053, a house edge of about 5.3%.
- Extended warranty: a $150 warranty that covers a $600 repair with a 15% chance has an expected payout of $90. Its expected value for the buyer is −$60.
- Business decision: a project that earns $500,000 with 40% probability and loses $200,000 with 60% has an expected value of $80,000.
Expected value is not the whole story
- Risk: two choices with the same expected value can carry very different risk. The standard deviation measures it.
- Ruin: a bet with a positive expected value can still bankrupt you if one loss is too large to absorb. Insurance makes sense despite its negative expected value because it protects against ruinous losses.
- One-off decisions: the expected value describes the average over many repetitions. For a single decision, the range of outcomes matters too.
Expected value in decision theory
Economists often replace money with utility, which captures the fact that the second $1 million matters less than the first. Maximizing expected utility explains why people buy insurance and diversify investments.