Expected value (EV) is the probability-weighted average of all possible outcomes. It tells you what you should expect to get on average if you repeat an event many times.
The Formula
E(X) = Σ (probability × value)
Sum all (probability × outcome) pairs.
Simple Example: Coin Flip Bet
A fair coin flip:
- Heads: win £10
- Tails: lose £8
E(X) = (0.5 × 10) + (0.5 × −8)
E(X) = 5 + (−4) = £1
Interpretation: On average, you win £1 per flip. This is a positive EV bet worth taking repeatedly.
Example: Insurance
Should you buy a £200/year phone insurance policy?
Assume:
- 5% chance of phone damage (cost: £400 to repair)
- 95% chance of no damage
Expected cost without insurance:
E(cost) = (0.05 × £400) + (0.95 × £0) = £20
Insurance cost: £200
The insurance costs £200 for expected damage of £20 — you're paying 10× the expected cost. Mathematically, insurance is a negative EV decision. However, the risk reduction may be worth the premium if you can't afford the £400 loss.
Gambling: The House Edge
A European roulette wheel (37 numbers, 0–36). You bet £1 on a single number:
- Win: 1 chance in 37, payout = £36 (35:1 + your stake)
- Lose: 36 chances in 37
E(X) = (1/37 × 36) + (36/37 × −1)
E(X) = 0.973 − 0.973 = −0.027
Expected loss = £0.027 per £1 bet = 2.7% house edge.
Over 1,000 spins of £1 each:
Expected loss = 1,000 × 0.027 = £27
Business Decision Making
A company is deciding whether to launch a product:
| Outcome | Probability | Profit/Loss |
|---|---|---|
| Strong success | 20% | +£500,000 |
| Moderate success | 40% | +£100,000 |
| Break even | 25% | £0 |
| Failure | 15% | −£200,000 |
EV = (0.2 × 500,000) + (0.4 × 100,000) + (0.25 × 0) + (0.15 × −200,000)
EV = 100,000 + 40,000 + 0 − 30,000 = £110,000
Positive EV → proceed with the project.
Limitations of Expected Value
- Variance matters: A 50% chance of winning £200 and a certain £100 have the same EV, but very different risk profiles
- Single events: EV only guarantees average outcomes over many repetitions
- Utility vs money: People value money non-linearly (risk aversion). Losing £1,000 hurts more than gaining £1,000 helps.
EV in Poker
Professional poker involves calculating pot odds and expected value for every decision. If a bet has positive EV, it should be made regardless of the short-term outcome.
EV = (probability of winning × pot size) − (probability of losing × bet size)