Expected value and edge in betting, with worked examples
Expected value is what a bet returns on average if it were repeated many times. How to work out EV and edge from a price and a probability.
Expected value (EV) answers one question: if you placed the same bet at the same price many times, what would you win or lose on average per bet? A price can be higher or lower than an outcome is worth, and EV turns that gap into a number.
This guide covers the formula, the related idea of edge, and three worked examples. It shows how to measure a price, not which bets to make.
The formula
You need two things: the price and a probability for the outcome.
EV = (chance of winning × profit if it wins) − (chance of losing × stake)
With decimal odds, the profit on a winning bet is stake × (odds − 1). Everything is per bet, averaged over the long run.
Edge is the same thing as a share of the stake:
edge = probability × decimal odds − 1
A positive edge means the price pays more than the probability says the outcome is worth. A negative edge means it pays less. An edge of zero is a fair price. Put another way, the price breaks even when the probability equals its implied probability, 1 ÷ odds.
Both formulas assume the bet either wins or loses. Where a push or void returns the stake (a whole-number spread or total, or draw no bet on a draw), add the chance of a push: edge = P(win) × odds + P(push) − 1. Void rules, for example for a player who does not play, vary by sportsbook.
Example 1: a coin flip at -110
Two even teams are each priced -110, which is decimal 1.9091. Say each has a 50% chance. On a stake of 100:
- Win: profit 100 × 0.90909 = 90.909, with a 50% chance, so 0.50 × 90.909 = 45.45
- Lose: 100, with a 50% chance, so 0.50 × 100 = 50.00
- EV: 45.45 − 50.00 = −4.55 per 100 staked
Edge: 0.50 × 1.9091 − 1 = −4.55%. That expected loss is the sportsbook's margin at work; the what the vig is guide explains where it comes from.
Example 2: a price above the fair price
The fair odds guide builds a consensus for a two-way market. Team B's fair probability comes out at 48.05%, a fair price of 2.081. One sportsbook offers 2.10. On a stake of 100:
- Win: profit 110, so 0.4805 × 110 = 52.86
- Lose: 100, so 0.5195 × 100 = 51.95
- EV: 52.86 − 51.95 = +0.91 per 100 staked
Edge: 0.4805 × 2.10 − 1 = +0.91%. The break-even probability at 2.10 is 1 ÷ 2.10 = 47.62%, a little below the fair 48.05%, which is why the result is positive.
A gap that small is easy to lose. If the same 2.10 is on an exchange that charges 2% commission on net winnings, the profit becomes 1.10 × 0.98 = 1.078, so the price is really 2.078. Then 0.4805 × 2.078 − 1 = about −0.15%: below fair once the fee is counted. Always use the price you would actually be paid. The betting exchanges guide covers fees.
Example 3: a bigger price
Take the three-way football (soccer) market from the fair odds guide, where one sportsbook prices the home win, draw and away win at 2.10, 3.40 and 3.60. Removing its margin gives the draw a fair probability of 28.06% (a fair price of about 3.56), a rougher estimate than a consensus. A different sportsbook offers 3.75 on the draw. On a stake of 100:
| Result | Chance | Profit | Chance × profit |
|---|---|---|---|
| Draw | 28.06% | +275.00 | +77.17 |
| Not a draw | 71.94% | −100.00 | −71.94 |
| EV | +5.23 |
Edge: 0.2806 × 3.75 − 1 = about +5.2%, a much bigger gap than example 2. Yet this bet loses almost three times in four. Over 100 such bets, the chance of at least one run of ten losses in a row is about two in three. The +5.23 is a long-run average; the short run can look very different.
Try your own price and probability:
Read as decimal odds: 2.10
Read as decimal odds: 2.00
Expected value +5.00 on a stake of 100.00 (+5.0% edge).
An average over many bets like this one, never a forecast of one result. It is only as good as the estimate.
Where the probability comes from
EV is only as good as the probability you put in. Nobody knows the true chance of an outcome, so every EV figure rests on an estimate.
- A consensus fair price is a common starting point: the median price across the sportsbooks that price a market, with the margin removed. OddsKeepr shows one for most markets, and the fair odds guide explains how it is built. The fewer sportsbooks behind it, the rougher the estimate.
- The margin method matters for long shots. OddsKeepr uses the proportional method; others give slightly different fair prices for big underdogs, enough to turn a small edge into none (see fair odds).
- Prices move. A gap seen early may close before the start. Comparing your price with the final price before the start is what closing line value measures.
- Small edges are fragile. At 2.10, each percentage point of probability is worth 2.1 points of edge, so an error of one point in the estimate can more than wipe out an edge of 1%.
EV and stake size
EV says whether a price is above or below an estimate of its worth. It does not say how much to stake. Staking too much on a positive-EV price can still lead to heavy losses through ordinary bad runs. The Kelly criterion is one way of relating stake size to edge, with its own limits.
Betting always carries a risk of losing money, so if you bet, set a limit you are comfortable with and check that the sportsbook is licensed where you are. If gambling stops feeling like entertainment, free and confidential help is listed for every country under where to get help.