What is an expected value calculator?
An expected value (EV) calculator shows what a bet is worth on average. It multiplies each possible result by its chance and adds the results: EV = chance of winning x profit if it wins – chance of losing x stake. At +150 with a 45% chance of winning, a bet of 100 units has an EV of +12.50 units. That is an average across many bets, not a prediction for any one bet.
The calculator needs four things: the stake, the odds, how often you think the bet wins, and how many bets you plan to place. The first two are facts. The third is an estimate, and it decides the whole answer. A bet with a positive EV is only worth making if that estimate is close to right, so the page spends time on how far results can stray from the average and where the estimate comes from.
How do you use the expected value calculator?
Type the stake, the odds in any format (decimal 2.50, fractional 3/2 or American +150) and your estimated chance of winning as a percentage. Add the number of bets you plan to make. The result updates as you type, runs in your browser and stores nothing.
- Enter the stake. This is the amount you risk on one bet, in units.
- Enter the odds you would get. Use the exact price, not an average.
- Enter your chance of winning. Use a percentage between 0.1 and 99.9. Be honest here, because the calculator trusts you.
- Enter the number of bets. It starts at 100.
- Read the large number. It is the EV of one bet. The panel below shows the break-even chance and the spread of results over your number of bets.
Example: Stake 100, odds +150 (decimal 2.50), chance 45%, 100 bets. A win pays 150 profit and a loss costs 100. EV = 0.45 x 150 – 0.55 x 100 = 67.50 – 55.00 = +12.50 per bet, which is +12.5% of the stake. Over 100 bets the expected result is +1,250 units.
How is expected value calculated?
Convert the odds to decimal odds d. A win then pays a profit of stake x (d – 1), and a loss costs the stake. With a win chance p, EV = p x stake x (d – 1) – (1 – p) x stake. The break-even chance is 1 / d: at that chance, EV is exactly zero. The calculator also finds the standard deviation of one bet, which is the square root of p x profit squared plus (1 – p) x stake squared, minus EV squared.
| Step | Working | Result |
|---|---|---|
| Decimal odds | 1 + 150 / 100 | 2.50 |
| Profit if the bet wins | 100 x (2.50 – 1) | 150.00 |
| Chance-weighted win | 0.45 x 150.00 | 67.50 |
| Chance-weighted loss | 0.55 x 100.00 | 55.00 |
| Expected value | 67.50 – 55.00 | +12.50 |
| Break-even chance | 100 / 2.50 | 40.00% |
For a casino game the chance is fixed by the rules, so EV is a known negative number. A 100-unit bet on a single number in double-zero roulette pays 35 to 1 (decimal 36.00) and wins with a chance of 1 in 38, or 2.63%. EV = (1/38) x 3,500 – (37/38) x 100 = 92.11 – 97.37 = -5.26 units, which is -5.26% of the stake. The roulette odds and payouts guide shows where that edge comes from, and Wikipedia gives the same figure as -1/19 per unit.
How much does your estimate change the answer?
The price stays the same in the table below (decimal 2.50, stake 100). Only the chance changes. A five-point change in the estimate moves the EV by 12.50 units each time, because each point of chance is worth 2.50 units at this price. The bet flips from a loss to a gain at 40%, the break-even chance.
| Your estimated chance | EV per 100 staked | EV as a share of the stake |
|---|---|---|
| 35% | -12.50 | -12.5% |
| 40% | 0.00 | +0.0% |
| 45% | 12.50 | +12.5% |
| 50% | 25.00 | +25.0% |
| 55% | 37.50 | +37.5% |
If you think the chance is 45% but it is really 40%, the “positive” bet has an EV of zero. If it is really 35%, the bet loses 12.50 units a time on average. A small estimation error can erase an edge, which is why EV calculators are better at showing what you would need to believe than at proving a bet is good.
Why can a bet with positive EV still lose money?
EV is an average. Any single bet either wins or loses, and a run of bets has a spread of outcomes around the average. The spread shrinks relative to the total as the number of bets grows, which is the law of large numbers, but it does not vanish. The table uses the default bet (stake 100, odds 2.50, chance 45%, EV +12.50) and counts every possible number of wins exactly.
| Number of bets | Expected result | Chance of finishing behind | Chance of finishing ahead |
|---|---|---|---|
| 1 | 12.50 | 55.0% | 45.0% |
| 10 | 125.00 | 26.6% | 49.6% |
| 50 | 625.00 | 19.7% | 71.4% |
| 100 | 1,250.00 | 13.4% | 81.7% |
| 500 | 6,250.00 | 1.1% | 98.6% |
| 1,000 | 12,500.00 | 0.1% | 99.9% |
After 100 bets the expected result is +1,250 units, but one standard deviation is about 1,244 units, so the expected gain is roughly one standard deviation wide. The calculator reports a 13.4% chance of finishing behind even though every bet has a positive EV. After 1,000 bets that chance is much smaller. This is why bankroll size matters, a point the risk of ruin calculator covers.
Where does your win chance come from?
The weak link in any EV calculation is the chance. There are three common sources. The first is a rule-based chance, such as the 1 in 38 for a roulette number, which is exact. The second is the chance implied by the odds, 1 divided by the decimal odds, which still contains the bookmaker margin and so overstates every outcome. The third is your own model, which can be right or wrong and is hard to check. Researchers who study betting markets find that margins mean that bets lose on average more than headline figures suggest, which is why an implied chance is not a fair chance.
If you use the price itself as your chance, the calculator will show an EV of zero or slightly negative every time, because the price is the break-even chance. A positive result needs a reason to believe the market is wrong, and that reason has to come from outside the calculator.
What does the calculator leave out?
- Margin and fees. It does not take commission, withdrawal fees or tax out of winnings.
- Stake limits. Real bets can be capped, cut or voided.
- Correlated bets. It treats the bets as independent. Two bets on the same game are not.
- Changing prices. It uses one price for every bet. Prices move.
- Bonuses and promotions. Their conditions change the maths and are not modelled.
- Your estimate. It cannot tell you if the chance you typed is realistic.
What mistakes do people make with expected value?
- Reading positive EV as a promise. It is an average. The table above shows the chance of finishing behind.
- Using the implied chance as the real chance. That bakes the margin into the answer and returns a result near zero or below.
- Using too few bets. With 10 bets the result is mostly luck.
- Forgetting that stakes grow. Raising the stake after a loss to win it back does not change the EV of each bet. It only changes how large the swings are.
- Mixing odds formats. Type a minus sign on negative American odds, so -110 is not read as 110.
What should you try next?
Try changing only the chance and watch the sign of the EV flip at the break-even chance. Then raise the number of bets to see the chance of finishing behind fall. If you want to see how much of a bankroll a losing streak can take, use the risk of ruin calculator. If you want to set limits first, the Safer play toolkit is the place to start.
Related Chancepedia tools are planned to sit alongside this page: the implied probability calculator, the break-even win rate calculator, the Kelly criterion calculator and the risk of ruin calculator. They will link here as they go live.
How to read the result
The large number is the expected value (EV) of one bet in units: the average result per bet if you could place it very many times. A positive EV means the bet gains on average, a negative EV means it loses on average. The line under it compares your chance with the break-even chance, which is 100 divided by the decimal odds. The panel shows the expected result over your number of bets, a typical swing (one standard deviation), and the exact chance of finishing behind, ahead or level. A positive EV can still finish behind.
How it is calculated
Your odds are first turned into decimal odds d. The profit on a win is stake x (d - 1). Expected value = chance x stake x (d - 1) - (1 - chance) x stake. The break-even chance is 100 / d. The standard deviation of one bet is the square root of chance x profit squared + (1 - chance) x stake squared - EV squared, and over n bets it is that figure times the square root of n. The chance of finishing behind is the sum of the exact binomial chances of every number of wins k where k x d is less than n.
Worked example
Worked example
Stake 100 at +150 (decimal 2.50) with a 45% chance. A win pays 100 x 1.50 = 150. EV = 0.45 x 150 - 0.55 x 100 = 67.50 - 55.00 = +12.50, which is 12.5% of the stake. Break-even chance: 100 / 2.50 = 40%. Over 100 bets the expected result is +1,250, one standard deviation is about 1,244, and the exact chance of finishing behind is 13.4%, ahead 81.7%, level 4.9%.
Limits
The calculator trusts the chance you enter. It does not know whether your estimate is realistic, and a few points of error can remove an edge. It treats every bet as independent and at the same price, so it does not model correlated bets, changing prices, stake limits, voided bets, bonuses, commission or tax. Fractional and American odds are converted exactly, but a rounded decimal price differs slightly from the same price in another format. Over a small number of bets the result is mostly luck. Results are estimates, not predictions.