The expected value formula for sports bets

The expected value of a bet is EV = p × profit − (1 − p) × stake, where p is the true chance the bet wins, profit is what it pays if it wins, and stake is what you lose if it does not. Divide by the stake and you get EV as a percentage, which in decimal odds is simply p × d − 1. A bet with positive EV is one where the price pays more than the outcome is worth; the number tells you how much you make per bet on average, not what will happen on this one.

Where p comes from

Everything in the formula is known except p. The odds tell you the profit; you choose the stake. The probability has to be estimated, and in practice that means a fair price: a sharp market's odds with the bookmaker's margin taken out. The devig guide shows how that removal works and why different methods give slightly different p. Your EV can never be more accurate than that estimate, which is the single most important caveat on this page.

The EV of a promotion

Promotions use the same formula with a different payout. A profit boost raises the profit term. A bonus bet removes the stake from the loss term, because losing it costs you nothing you paid for, and removes it from the win too, because the stake is not returned. That is why promos can make an otherwise ordinary bet strongly positive, and why profit boosts and bonus bets are usually the first +EV bets people ever place.

EV is an average, not a prediction

No single bet ever returns its EV. The −110 bet above pays +$100 or −$110; it never pays $3.40. EV is what the bet returns per play averaged over a very large number of identical plays, and it only shows up in your balance once you have placed enough of them for the wins and losses to even out. How many that takes is the subject of the variance guide: at a 3% edge, it is usually thousands, not dozens. Until then, closing line value tells you far sooner whether your prices were good.

Adding up EV across bets

Expected values add. Thirty bets a week at $50 and a 3% average edge is 30 × $50 × 0.03 = $45 a week expected, whatever the individual results. That is the right way to judge a strategy before you start: total the edges you can realistically find and place, multiply by the stakes your books will accept, and see whether the answer justifies the time. The positive EV guide covers why the result still feels like losing for long stretches.

Letting the board do the arithmetic

The Positive EV board runs this formula on every price it collects: it builds a fair probability for each market, compares every book's price against it, and lists the ones where p × d − 1 is above your minimum edge, net of venue fees on exchange rows. The walkthrough works one row through by hand.