Expected Value
The probability-weighted average outcome of a bet, expressed as a per-unit dollar return. The single number that tells you whether a wager is profitable in the long run.
Expected value is the fundamental concept underlying every bet-sizing decision. For a binary wager at decimal odds d with estimated win probability p, the expected return per dollar staked is (p × d) − 1. Positive EV bets are profitable in expectation; negative EV bets are not. Realized outcome can differ wildly from EV on any single bet, but across a sufficient number of bets the realized result converges on the expected value, which is why bet selection on EV terms is the only durable framing for serious bettors.
- Bet selection on any other criterion (gut, hot streak, narrative, vibes) does not converge to profitability. EV is the only filter that is mathematically guaranteed to produce positive long-run returns when applied honestly.
- Kelly stake sizing requires a probability and an EV; it is the optimal sizing rule for translating EV into a bet size. Without an estimated EV, no sizing rule is principled.
- EV decomposition reveals where a model's edge actually lives: which sports, which markets, which bet types, which odds ranges. Aggregate EV often masks high-EV pockets that are smaller than the noise of negative-EV regions.
- For institutional buyers, every model output is graded on EV terms. A 70%-accurate model that produces small positive EV is often more valuable than an 80%-accurate model with negative EV after vig.
EV = (p × decimal_odds) − 1Multiply your estimated win probability by the decimal odds, then subtract 1. The result is the expected return per dollar staked. Multiply by stake to get the dollar EV of a specific bet. Negative EV means the bet is unprofitable in expectation.
You estimate a 55% win probability on a bet at +110 (decimal 2.10). EV = 0.55 × 2.10 − 1 = +0.155, or +15.5% per dollar staked. On a $100 bet, EV is +$15.50. If you actually win, you net +$110; if you lose, you net -$100. Across many bets at this same EV, the average return per bet converges to +$15.50.
- Confusing realized outcome with expected value. A losing bet can have been positive EV; a winning bet can have been negative EV. The math only converges over many trials.
- Computing EV against raw implied probability instead of no-vig fair probability when comparing against the market. EV calculations should be expressed in real fair-odds terms.
- Ignoring CLV when measuring whether claimed EV is real. Backtest EV that doesn't survive an empirical CLV haircut is not real EV; it's an artifact of the difference between live odds and reference odds.
Compute it yourself.
We built a free calculator that implements the formula above. Plug in your numbers and see the math.
Open Kelly Stake CalculatorWhat does positive EV mean in concrete terms?
A bet with positive EV returns more, on average, than it costs to place. A +5% EV bet placed 1,000 times at consistent stakes will tend to produce a 5% profit on total wagered. Single-bet variance can push the realized outcome anywhere; aggregate variance shrinks with sample size.
How do I find positive EV bets?
Either build or license a calibrated probability model that estimates true outcome rates, then compare those probabilities to the no-vig market price. When your probability exceeds the no-vig implied probability by a meaningful margin, the bet is positive EV. Most retail bettors do not actually do this; they bet on narrative.
Is positive EV the same as positive ROI?
Over a long enough sample, yes. On any short sample, no, variance dominates. Sharp shops use Beta-Binomial confidence intervals on win rate to know how many bets they need before realized ROI is statistically distinguishable from zero. The answer is usually larger than people expect.
How does EV relate to Kelly sizing?
Kelly is the sizing rule that maximizes long-run wealth growth given an EV estimate. The Kelly fraction is mathematically derived from p and the decimal odds, but the input is exactly the EV calculation: Kelly only recommends a positive stake when EV is positive. Negative EV always returns a negative Kelly fraction, which is interpreted as 'do not bet.'