Expected value (EV) is the long-run average profit or loss per unit staked, found by comparing your estimated probability with the price's implied probability. The key idea is that EV is not a prediction of the next result; it is a measurement that is only as reliable as the probability estimate and the odds you actually record. This article is educational analysis, not betting advice.
Formula
For decimal odds \(O\) and an estimated probability \(p\), the EV per 1 unit staked is:
\[ EV = p \times (O - 1) - (1 - p) \times 1 \]
which simplifies to:
\[ EV = pO - 1 \]
For a stake \(S\):
\[ EV = S \times (pO - 1) \]
A calculator only needs three inputs: \(p\), decimal odds \(O\), and stake \(S\). If you use American odds, convert to decimal first. For positive American odds \(A\), decimal odds are \(1 + A/100\). For negative American odds \(A\), decimal odds are \(1 + 100/|A|\).
The break-even probability is:
\[ p_{break-even} = \frac{1}{O} \]
The edge over implied probability is:
\[ edge = p - \frac{1}{O} \]
If \(O = 2.60\), the break-even probability is \(1/2.60 = 38.46\%\). EV is positive only when your estimated probability is above that break-even mark.
Worked example
Take a recent settled fixture in AlphaBench data: Khovd Western vs Deren (Premier League). Suppose, purely for illustration, the decimal price for a Khovd Western win is 2.60. AlphaBench tracks models such as deepseek-flash (DeepSeek), gpt-5-mini (OpenAI), and grok-build-0.1 (xAI). Imagine those models produced the following hypothetical probabilities for the home win:
| Model | Hypothetical \(p\) | EV per 1 unit | EV per 100 units | |---|---:|---:|---:| | deepseek-flash | 42% | \(0.42 \times 2.60 - 1 = +0.092\) | +9.2 | | gpt-5-mini | 45% | \(0.45 \times 2.60 - 1 = +0.170\) | +17.0 | | grok-build-0.1 | 38% | \(0.38 \times 2.60 - 1 = -0.012\) | -1.2 |
Under these assumptions, the first two probabilities imply positive EV, while the third is slightly negative. The implied probability from the odds is \(1/2.60 = 38.46\%\), so only probabilities above 38.46% produce positive EV at that price. These are illustrative numbers, not actual AlphaBench predictions or recommendations.
Relationship to implied probability
EV is a direct comparison between your probability and the market's implied probability. Decimal odds \(O\) imply probability \(1/O\). If your model says 45% and the odds imply 38.46%, the gap is the claimed edge. If your model says 38%, there is no positive EV at those odds.
Implied probability usually includes a bookmaker margin, so the implied probabilities across all outcomes often sum to more than 100%. For that reason, serious evaluation often compares model probabilities with no-vig or closing implied probabilities. Closing line value (CLV) tracks whether the price obtained was better than the closing price, which is often treated as a sharper estimate of the market's final view.
EV also depends on calibration, not just discrimination. A model can rank outcomes well but still be miscalibrated. If it says 45% too often when the true rate is 38%, EV estimates will be overstated. Metrics such as Brier score and log loss help evaluate probability accuracy, while calibration checks whether events assigned 45% actually occur about 45% of the time. See methodology and Brier score for football predictions for more on these ideas.
Variance and sample size
EV is an average, not a guarantee. For a binary bet with decimal odds \(O\) and probability \(p\), the variance per unit staked is:
\[ Var = p(1-p)O^2 \]
The standard deviation is:
\[ SD = O\sqrt{p(1-p)} \]
For example, with \(p = 0.45\) and \(O = 2.60\):
\[ EV = 0.45 \times 2.60 - 1 = +0.17 \]
\[ Var = 0.45 \times 0.55 \times 2.60^2 \approx 1.673 \]
\[ SD \approx 1.293 \]
So the EV per unit is +0.17, but the single-bet standard deviation is about 1.293 units. Over \(N\) independent bets with the same profile, total EV is roughly \(N \times 0.17\), while total standard deviation is roughly \(\sqrt{N} \times 1.293\). A rough 95% confidence calculation for a positive result would require hundreds of such bets, and real samples are rarely this clean or independent.
This is why ROI, maximum drawdown, and sample size matter. A positive EV estimate can experience long losing runs. A small sample can look profitable or unprofitable by chance. AlphaBench evaluates models over many fixtures and scoring metrics rather than relying on one result; see how AlphaBench scores models and the leaderboard.
Common mistakes
- Confusing EV with win probability. A 90% chance at very short odds can still have negative EV. - Treating implied probability as the true probability. It includes margin and market noise. - Ignoring overround. Summing implied probabilities across outcomes often exceeds 100%. - Using uncalibrated model probabilities. Positive EV calculations collapse if \(p\) is wrong. - Judging an EV estimate after one result. Variance can dominate a single outcome. - Overfitting backtests. A model that looks good in-sample may not be calibrated out-of-sample. - Ignoring execution details such as odds movement, limits, liquidity, and costs. - Comparing EV across correlated or non-independent opportunities as if they were identical. - Assuming Brier score or log loss directly measure profit. They measure probability accuracy, not returns. - Forgetting that positive EV bets can lose and negative EV bets can win. EV is about the long-run mean.
Limitations
EV is model-dependent. The true probability is unknown, so every EV number is an estimate. Markets are often efficient, meaning apparent edges may be small, stale, or illusory. Odds change quickly, and closing odds are usually a stronger benchmark than earlier prices. Historical data can be non-stationary, so a model that worked in one period may not work in another.
EV also ignores risk of ruin, stake sizing, correlation, taxes, fees, limits, and personal utility. A positive EV opportunity can still be unavailable at scale, or can produce drawdowns that are difficult to tolerate. AlphaBench uses EV concepts for evaluation and education, not as a recommendation to place wagers. Nothing here is betting advice.
Related reading
- AlphaBench methodology - Brier score for football predictions - How AlphaBench scores models - AlphaBench leaderboard