Expected value is a long-run average under probability assumptions; a positive estimate does not prevent an individual loss.
The expected value formula
Expected value (EV) is the average result of a bet if it could be repeated many times under the same conditions. For a simple win-or-lose bet:
EV = P(win) × profit if it wins − P(loss) × stake
Take a $100 bet at +110 with a 50% chance to win: 0.50 × $110 − 0.50 × $100 = +$5. On average, each bet like it earns $5, or 5% of the stake, even though no single bet ever pays $5. Each one either wins $110 or loses $100.
The break-even probability at +110 is 100 ÷ 210 = 47.62%. Any estimate above that makes the bet +EV, and any estimate below it makes the bet −EV.
A worked example with a market reference
The 50% has to come from somewhere. One common source is the no-vig price from a widely bet market. Suppose the main market prices a game at −125/+105:
- Implied probabilities: −125 is 55.56% and +105 is 48.78%, a total of 104.34%.
- No-vig probabilities: 53.25% for the favorite and 46.75% for the underdog.
- Another book offers the underdog at +120, which has a break-even rate of 45.45%.
- EV per $100: 0.46753 × $120 − 0.53247 × $100 = +$2.86, about 2.9% of the stake.
The edge is small, and it depends on the main market being close to fair. Try the numbers yourself below: enter +120 and a 47% estimate, then lower the estimate until the expected value turns negative.
Why +EV bets still lose often
That +120 bet loses about 53% of the time. A single $100 bet has a standard deviation of about $110 against an expected gain under $3, so short-run results are mostly noise. Over 1,000 such bets the expected profit is about $2,860, but the standard deviation is about $3,470, which leaves roughly a one-in-five chance of being behind after all 1,000 bets.
That is why +EV betting is judged on process rather than short-run results, and why stakes have to be small enough to survive long losing stretches. The bankroll management guide covers sizing.
The estimate carries the risk
EV is only as accurate as the probability behind it. In the example, if the underdog’s true chance is 45% rather than 46.75%, the same +120 bet is worth −1.0% of the stake: an error of less than two percentage points erases the edge. Record each probability and its source before the event, and check over time whether your prices beat the closing line.
Pushes, voids, stake limits, and correlated bets all change the arithmetic, and a price that disappears before you can bet it has no expected value at all. Positive EV describes a price and an estimate; it is not a promise of profit.
Does this topic predict the next result?
No. A definition, historical statistic, market price, or model output is not a guarantee. Treat estimates as conditional on their data and assumptions, and keep the uncertainty visible.