Every bet has two numbers attached to it. One is the price the book is offering. The other is the real chance the bet wins. Expected value, or EV, is what you get when you put those two together: the amount you'd win or lose per bet, on average, if you could make the same bet thousands of times.
A bet with positive EV pays you more than its risk deserves. A bet with negative EV pays you less. Most bets at most books are slightly negative, because the book's margin is built into the price. The whole job of a disciplined bettor is to find the ones that aren't.
The formula
EV = (chance of winning × profit if you win) − (chance of losing × stake)
It's easiest to work per $1 staked, which gives you EV as a percentage of your stake. For American odds, the profit on $1 is:
- Positive odds: odds ÷ 100. So +120 profits $1.20.
- Negative odds: 100 ÷ |odds|. So -150 profits $0.667.
Three worked examples
+120 at a true 50% chance. You win $1.20 half the time and lose $1 half the time.
(0.50 × 1.20) − (0.50 × 1.00) = 0.60 − 0.50 = +0.10
That's +10% EV, or +$10 per $100 bet. The break-even win rate for +120 is 100 ÷ 220 = 45.45%, and you're well above it.
+150 at a true 45% chance. This one wins less often than it loses, but the payout makes up for it.
(0.45 × 1.50) − (0.55 × 1.00) = 0.675 − 0.55 = +0.125
That's +12.5% EV. +150 only needs 40% to break even, so 45% is a comfortable edge. Underdogs aren't bad bets just because they lose more often. The price is what matters.
-150 at a true 55% chance. This one wins more often than not, and it's still a bad bet.
(0.55 × 0.667) − (0.45 × 1.00) = 0.367 − 0.45 = −0.083
That's about −8.3% EV. -150 needs 60% to break even. A 55% favorite priced like a 60% favorite is a losing bet, no matter how confident it feels. The EV calculator runs this for any price and probability.
Where the true chance comes from
The formula is simple. The hard part is the "true chance" input. Nobody hands it to you. There are two common ways to get a reasonable estimate.
De-vig a sharp two-way market. A book's prices include its margin, so the raw implied probabilities add up to more than 100%. Strip the margin out and you get that book's fair estimate. Say a sharp book has a prop at over -125, under +105:
- -125 implies 55.56%. +105 implies 48.78%. Total: 104.34%.
- Fair over: 55.56 ÷ 104.34 = 53.25%. Fair under: 46.75%.
If another book is offering the under at +120, the EV is (0.4675 × 1.20) − 0.5325 = +2.86%. The no-vig calculator does the de-vig step for you, and our post on no-vig fair odds walks through it in detail.
This approach borrows the sharp book's opinion. It works best on main lines at books that move quickly, and less well on far alt lines or numbers that haven't caught up to news.
Your own model. If you project player stats yourself, your model's probability can be the input. This is harder to get right than it sounds. A model that's off by a few percentage points will show you edges that aren't there. If your numbers regularly disagree with every sharp book by a wide margin, the books are usually the ones that are right.
Why +EV bets still lose
A +10% EV bet at +120 still loses half the time. EV describes the long-run average, not the next result.
Here's what that looks like in practice. Take the +120 bet at a true 50%, $100 a bet, 100 times:
- Expected profit: +$1,000.
- Standard deviation of that result: about $1,100.
The spread is bigger than the edge. Over those 100 bets, there's roughly an 18% chance you finish in the red, even though every single bet was +10% EV. Over 50 bets, it's about 24%.
Losing streaks are normal too. At a 50% win rate, the chance of hitting at least one run of five straight losses somewhere in 100 bets is about 81%. A run of seven or more happens about 32% of the time.
None of that means the bets were bad. It means 100 bets is a small sample.
Variance and what to do about it
Variance is the gap between what should happen and what does. You can't remove it, but you can live with it:
- Size bets so a bad run doesn't end you. A fixed small percentage of bankroll per bet, or a fraction of Kelly, keeps a normal losing streak from becoming a crisis.
- Judge the process, not the last week. Track whether you're getting better prices than the market settles on. Closing line value gives you a signal much faster than win-loss does.
- Keep the edge honest. Small, real edges beat large, imagined ones. Every percentage point you overstate your probability by comes straight out of your EV.
The practical takeaway
Before placing a bet, ask two questions: what's the fair chance this wins, and what price am I being offered? If the price implies a lower chance than your fair estimate, the bet is +EV. If not, pass, however good the pick looks.
You don't need a model to start. De-vig a sharp book's price with the no-vig calculator, compare it to the best price available elsewhere, and plug both into the EV calculator. Then expect to lose plenty of those bets anyway, and keep making them.