Formula for a one-unit back bet
Let p be estimated win probability and d be decimal odds. A winning one-unit bet makes d - 1 units of profit; a loss loses one unit.
EV = p * (d - 1) - (1 - p)
The same expression simplifies to EV = (p * d) - 1. OpenStax's expected-value chapter supports weighting each possible payoff by its probability.
Worked example: one price
For p = 0.46 and d = 2.30:
- Win profit = 2.30 - 1 = 1.30 units.
- Loss probability = 1 - 0.46 = 0.54.
- EV = (0.46 * 1.30) - (0.54 * 1) = 0.058 units.
- Equivalent EV per unit = (0.46 * 2.30) - 1 = 0.058 units.
The illustrative expectation is 5.8% of stake before costs and estimation error. It is not an expected profit for the next single bet.
Compare two available prices
Keep the probability fixed at 0.46:
| Decimal price | Break-even probability | EV per unit |
|---|---|---|
| 2.10 | 47.62% | -0.034 |
| 2.20 | 45.45% | 0.012 |
| 2.30 | 43.48% | 0.058 |
At 2.10, EV = (0.46 * 2.10) - 1 = -0.034 units. At 2.20, EV = (0.46 * 2.20) - 1 = 0.012 units. Price comparison changes the decision without changing the football forecast.
Test probability sensitivity
A point estimate hides uncertainty. At odds 2.30:
| Estimated p | EV calculation | EV |
|---|---|---|
| 0.42 | (0.42 * 2.30) - 1 | -0.034 |
| 0.44 | (0.44 * 2.30) - 1 | 0.012 |
| 0.46 | (0.46 * 2.30) - 1 | 0.058 |
| 0.48 | (0.48 * 2.30) - 1 | 0.104 |
If plausible model variation spans 42% to 46%, the sign is not stable. Record that ambiguity instead of presenting 5.8% as precise. Calibration guidance supports checking probability reliability rather than treating a model output as exact.
EV for non-binary settlement
For pushes, half wins, half losses, cash-outs or exchange commission, list every mutually exclusive outcome and its net profit. Then calculate:
EV = sum of (outcome probability * net outcome profit)
Do not force a quarter-handicap or promotion into a two-outcome formula. The probabilities must sum to one and the payoffs must reflect the current product rules.
Verify the inputs
- Confirm the market and settlement period.
- Convert the accepted price to decimal odds.
- Use the probability saved at the decision cutoff.
- Include commission or other known costs.
- Recalculate independently in a spreadsheet or tested function.
- Preserve the row even if the bet loses, is void, or is rejected.
Probability forecasts should also be checked for calibration on later events. scikit-learn's calibration guide explains why a collection of forecasts near p should resolve near that frequency when the model is reliable.
Verify the formula independently
Test the calculation with three boundary cases before trusting a spreadsheet or application. If p = 0, EV must be -1 for a one-unit binary back bet. If p = 1, EV must equal d - 1. At the break-even probability p = 1 / d, EV must equal zero apart from rounding.
For d = 2.50, break-even p = 1 / 2.50 = 0.40 and EV = (0.40 * 2.50) - 1 = 0. These checks catch sign errors, accidental use of gross return as profit, and percentage-versus-decimal mistakes. Preserve the test cases beside the production formula.
Continue learning
- Next guide: How to Audit Football Tipster Claims and Records
- Related guide: Line Shopping
- Definition: Expected Value
Assumptions and limitations
All examples are illustrative, use a one-unit stake, and exclude taxes and operator-specific settlement. EV is a model expectation, not a forecast of the next result. It does not determine an affordable stake; staking and drawdown are covered in the bankroll-management collection.
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