Expected Value (EV) in Betting: Formula and Worked Examples
The EV formula taken apart term by term, three bets settled in naira, the strike rate every price has to beat, and what changes when a handicap 0 gives the stake back.
Expected value is the average result of a bet if the same situation could be settled a thousand times. It has one formula: multiply what you win by how often you win it, subtract what you lose multiplied by how often you lose it. A ₦500 stake at 4.20 on an outcome you make 26% returns 0.26 × ₦1,600 − 0.74 × ₦500 = +₦46. That figure will never appear on a settled slip — the bet pays ₦2,100 or nothing — but it is the number that decides whether placing it was correct.
In short.
- EV = (chance of winning × profit) − (chance of losing × stake). Per naira staked it collapses to chance × price − 1.
- Positive EV means the price is longer than your opinion deserves; negative means you are paying over the odds for it.
- Every price carries a break-even strike rate: 55.6% at 1.80, 25% at 4.00. Below it the bet loses money by arithmetic.
- When a draw refunds the stake, the losing term shrinks and the formula needs a third line — otherwise handicap 0 looks far worse than it is.
- A probability taken from the odds themselves returns an EV of exactly zero. The estimate has to come from somewhere else.
The formula, term by term
Written out in full, expected value has four ingredients and no hidden ones.
| Term | What it is | Where you get it |
|---|---|---|
| Profit if it wins | Stake × (decimal price − 1) | The slip: ₦500 at 4.20 pays ₦1,600 of profit |
| Chance of winning | Your probability, as a decimal | Your own estimate — never the price |
| Loss if it loses | The stake | ₦500 |
| Chance of losing | 1 minus your probability | Follows from the estimate |
Multiply the first pair, multiply the second pair, subtract. Dividing the answer by the stake gives the same thing as a percentage, and the percentage has a shortcut worth memorising: your probability × the price, minus 1. At 4.20 and 26% that is 0.26 × 4.20 − 1 = +9.2%, and 9.2% of ₦500 is the ₦46 above.
The percentage form is the one to compare across bets: ₦46 expected on ₦500 and ₦46 on ₦2,000 are not the same quality of bet. Prices and percentages convert both ways in turning odds into probability.
Three slips settled on paper
Each uses the same ₦500 stake and the same two steps: the naira version first, then the percentage check.
Example 1 — a short favourite. A Premier League home side is 1.45 and you make them 72%. Profit if it wins: ₦500 × 0.45 = ₦225. EV = 0.72 × ₦225 − 0.28 × ₦500 = ₦162 − ₦140 = +₦22. Check: 0.72 × 1.45 − 1 = +4.4%, and 4.4% of ₦500 is ₦22.
Example 2 — an outsider. An away side in La Liga is 4.20 and you make them 26%. Profit if it wins: ₦500 × 3.20 = ₦1,600. EV = 0.26 × ₦1,600 − 0.74 × ₦500 = ₦416 − ₦370 = +₦46. Check: 0.26 × 4.20 − 1 = +9.2%.
Example 3 — a bet that fails the test. Over 2.5 goals is 1.80 and your scoring model says 53%. Profit if it wins: ₦500 × 0.80 = ₦400. EV = 0.53 × ₦400 − 0.47 × ₦500 = ₦212 − ₦235 = −₦23. Check: 0.53 × 1.80 − 1 = −4.6%. The opinion is that the goals are more likely than not; the price still makes backing it a mistake.
Example 3 is the useful one. A 53% call is a winning opinion in the ordinary sense — the outcome happens more often than not — and it is a losing bet at 1.80, because 1.80 demands 55.6%. Being right about what will probably happen is not the same as being paid enough for it.
The strike rate each price has to beat
Divide 1 by the price and you have the hit rate at which a bet exactly breaks even. The last column shows what a single extra win beyond that rate adds across 100 flat ₦500 bets: it is always ₦500 × the price, because one settlement swings from −₦500 to +₦500 × (price − 1).
| Price | Break-even strike rate | Strike rate needed for +5% EV | What one extra win in 100 is worth |
|---|---|---|---|
| 1.30 | 76.9% | 80.8% | ₦650 |
| 1.50 | 66.7% | 70.0% | ₦750 |
| 1.80 | 55.6% | 58.3% | ₦900 |
| 2.00 | 50.0% | 52.5% | ₦1,000 |
| 2.50 | 40.0% | 42.0% | ₦1,250 |
| 3.00 | 33.3% | 35.0% | ₦1,500 |
| 4.00 | 25.0% | 26.3% | ₦2,000 |
| 6.00 | 16.7% | 17.5% | ₦3,000 |
| 10.00 | 10.0% | 10.5% | ₦5,000 |
The middle column is the same 5% edge expressed nine ways, and it collapses as prices lengthen: 3.9 percentage points of accuracy at 1.30, half a point at 10.00. That asymmetry is why a 5% edge claimed on a long shot deserves far more scrutiny than the same claim on a favourite — half a point is inside the noise of any hand-built estimate.
When the stake comes back: EV with a refund
Handicap 0 — the same bet the app may label draw no bet — has three outcomes rather than two, and the middle one returns your money. The formula gains a term that is worth nothing and costs nothing, and the losing term shrinks accordingly.
Take a fixture you have priced at home 52%, draw 26%, away 22%, with handicap 0 on the home side offered at 1.50.
EV = 0.52 × ₦250 + 0.26 × ₦0 − 0.22 × ₦500 = ₦130 − ₦110 = +₦20 on a ₦500 stake, an expectation of +4.0%.
Two different percentages describe that bet, and both are correct. Measured against every naira staked it is +4.0%. Measured against the naira actually at risk — the 74% of the time the bet resolves — it is +5.4%, because 4.0 ÷ 0.74 = 5.4. Use the first when comparing with ordinary bets, the second when judging the quality of the read.
The common error here is applying the two-outcome shortcut and getting 0.52 × 1.50 − 1 = −22%, which would condemn a perfectly sound bet. The settlement rules for refunded stakes are set out in the draw no bet guide. For comparison, backing the same side in the ordinary result market at 1.95 gives 0.52 × 1.95 − 1 = +1.4%, or ₦7 — the cover is the better ticket on these numbers.
Why one settled bet says nothing about the decision
The 4.20 bet from Example 2 pays ₦2,100 or nothing. Its ₦46 expectation is an average over a distribution the bet never visits, which is why "it lost, so it was a bad bet" is a statement about luck rather than judgement.
Scale it up and the point sharpens. A hundred such bets carry an expected profit of ₦4,600 on ₦50,000 staked, but the typical spread around that figure is about ₦9,200 — twice the expectation. Three such runs in ten finish in the red, and the decision was right in every one of them.
Important. The reverse also holds, and is more dangerous. A negative-EV bet wins its share of the time, and winning is the strongest possible reinforcement for a habit that loses money. Judge slips by what the numbers were when you placed them, not by how they settled.
The error that makes every bet look positive
The formula returns whatever quality of probability you feed it, and there is one input that guarantees a wrong answer: a probability derived from the price you are testing.
Raw implied probability. At 2.10 the implied chance is 1 ÷ 2.10 = 47.62%. Put it back through the formula and 0.4762 × 2.10 − 1 = 0. Every bet on the board scores zero, which is arithmetically true and practically useless.
Margin-cleaned probability. Suppose the book is 2.10 / 3.40 / 3.80, implying 47.62% + 29.41% + 26.32% = 103.35%. Divide through and the home side cleans to 46.08%. Test that against the same 2.10 and you get 0.4608 × 2.10 − 1 = −3.2% — every bet on that book scores minus the mark-up, exactly as margin arithmetic predicts. Correct, and still not an estimate.
Implied probability with a nudge. The most common version: read 47.6% off the price, decide the home side is "a bit better than that", write 52%, and produce a 9% edge from nothing. Any figure produced after the price is visible is contaminated by it, which is why the estimate belongs in your notes before the board is open — the discipline set out in the value betting guide.
FAQ
Is a positive-EV bet the same thing as a good bet?
It is a necessary condition, not a sufficient one. A bet can carry positive expectation and still be wrong for you if the stake is too large for the bank, if you hold three correlated positions on the same match, or if the edge is thin enough to vanish under a small error in the estimate.
How do I calculate EV for an accumulator?
Multiply the per-naira figures rather than adding them. Two independent legs at +5% each give 1.05 × 1.05 − 1 = +10.25%. The catch is that the same multiplication punishes mistakes: pair a +5% leg with a leg that is really −5% and the ticket comes to −0.25%.
What EV should I be looking for on ordinary markets?
Something that survives being wrong. On main football markets an edge under about 4% is inside the error of most estimates and inside the rounding of the prices themselves. Treat 4–8% as a normal working range and anything far above it as a prompt to check for team news you have missed.
Does EV apply to bonuses and free bets?
Yes, with one change: a free bet returns profit only, so the losing term is zero. That makes the arithmetic look generous, but rollover conditions add turnover you would not otherwise have placed, and that turnover carries the ordinary negative expectation.
Read next
- Closing line value — checking whether your probabilities are any good without waiting for a season of results.
- The Kelly criterion — turning an expectation into a stake size.
- Today's predictions — fixtures to price before you look at the board.
This article is for information only and is not an inducement to gamble. Betting involves the risk of losing money — never stake more than you can afford to lose. 18+. If gambling stops being entertainment, read our responsible gambling guide and seek help.
