What is expected value in betting?
Expected value (EV) is the probability-weighted average of everything that can happen to your stake. For a simple bet with two outcomes, you win your profit with probability p and lose your stake with probability 1 − p:
EV = p × profit − (1 − p) × stake
Dividing by the stake gives a percentage, the expected return on every unit you bet:
EV % = odds × p − 1 (decimal odds)
Example: a bet at 2.10 that is worth 1.97
Suppose the fair odds of an outcome are 1.97, which means a probability of 1 ÷ 1.97 = 50.8%. Your bookmaker pays 2.10. You stake 100:
- If it wins (50.8% of the time) you profit 110.
- If it loses (49.2% of the time) you lose 100.
- EV = 0.508 × 110 − 0.492 × 100 = +6.6, or +6.6% of the stake.
Any single bet will either win 110 or lose 100: neither is 6.6. The number only describes the long run. You can check this one with the EV calculator.
How long is the long run?
That +6.6% edge at odds of 2.10 has a standard deviation of about 1.05 units per bet. Here is what it means for your results (approximate, from the normal distribution):
| Number of bets | Expected profit (1 unit each) | Typical swing (±1 st. dev.) | Chance of being ahead |
|---|---|---|---|
| 10 | +0.7 | ±3.3 | 58% |
| 100 | +6.6 | ±10.5 | 74% |
| 250 | +16.5 | ±16.6 | 84% |
| 500 | +33.0 | ±23.5 | 92% |
| 1,000 | +66.0 | ±33.2 | 98% |
| 2,000 | +132.0 | ±47.0 | 99.8% |
After ten bets, a +6.6% edge is barely better than a coin flip at being ahead. After a thousand, it almost always shows. That's why judging a value strategy over a week or a month is a mistake.
Where the probability comes from
EV is only as good as the probability you feed it. If you estimate 55% for something that is really 48%, a terrible bet looks like a bargain. Instead of guessing, value bettors use the price of the sharpest market with its margin removed (see how to find value bets) and then check the estimate with closing line value: if your prices keep beating the fair close, the probabilities are right.
+EV is not a prediction. It doesn't say the bet will win or that you are 'right' about the match. It says the price is better than the odds of winning justify.
EV and how much to stake
Two bets with the same EV% can deserve very different stakes: a long shot with +10% EV is far more volatile than a favourite with +3%. Staking in proportion to the edge is the idea behind the Kelly criterion; read flat betting vs Kelly and try the Kelly calculator.
Expected value with American odds
The formula is the same; only the way you read the price changes. At +110 a 100 stake profits 110. If the true probability is 50% (fair odds +100), EV = 0.50 × 110 − 0.50 × 100 = +5, which is +5% of the stake. The EV calculator takes American, decimal and fractional odds.
EV, ROI and yield: what's the difference?
- EV % is what you *expect* to make per unit staked on a bet or a group of bets, before they are played.
- ROI or yield is what you *actually* made per unit staked after they're played.
- Over a small sample ROI can be anywhere; over a large one it should approach the EV if your probabilities were right. The gap between the two is the variance you saw in the table above.
Accumulators and EV
The EV of an accumulator is the product of each leg's (probability × odds). If every leg has a +3% edge, three legs have an edge of about 1.03³ − 1 = +9.3%. But if each leg has a margin and a small error in its probability, the combination multiplies those too, which is why accumulators are so much riskier than their legs.
Common mistakes with expected value
Common mistakes with expected value
- Calling any bet you like +EV. If the probability is a hunch, so is the EV.
- Ignoring the margin. Comparing your price with the reference's raw price, which includes the bookmaker's fee, overstates the edge. Remove it first with the no-vig calculator.
- Judging by results. A +EV bet that lost was still a good bet; a −EV bet that won was still a bad one.
- Forgetting you can't take the price. An edge you can't place, or that has moved, is worth nothing.