Expected Value in Sports Betting

Every bet has two outcomes and a probability attached to each. Expected value weights the outcomes by those probabilities and adds them up. If the answer is positive, the bet makes money on average. If it is negative, it does not, and no amount of conviction changes that.

EV = p × stake × (decimal − 1) − (1 − p) × stake

  p       your probability estimate
  decimal the price on offer

Note what the formula requires: your own estimate of the probability. This is the whole difficulty of betting, compressed into one variable. The price is handed to you; the probability is not, and every EV figure you calculate is only as good as the number you put there.

Before any EV calculation, work out what the price demands. Divide 1 by the decimal odds and you have the implied probability — which is also the exact win rate at which the bet makes nothing.

That last column is the argument against ever quoting a win rate on its own. A 55% hit rate is a strong edge at −110 and a losing operation at −140. What matters is the gap between your probability and the one the price implies — nothing else.

A spread is priced at −110 and you believe the side is genuinely 55% to win. You are betting $100.

decimal(−110)   = 1.909
break-even      = 1 / 1.909 = 52.38%
edge            = 55% − 52.38% = 2.62 percentage points

EV = 0.55 × 100 × 0.909 − 0.45 × 100
   = 50.00 − 45.00
   = +$5.00 per $100 staked   (a 5.00% return on turnover)

Five dollars, on a bet that resolves as either +$90.91 or −$100.00. It is a real edge and a good one — professional sports bettors operate on considerably less — and it is completely invisible in any single result.

Move the estimate and watch how fast it collapses. At 53% the same bet is worth $1.18. At 52.38% it is worth nothing at all. At 50% — an honest coin flip — it is −$4.55, which is precisely the sportsbook’s hold. Two and a half points of probability, in a domain where nobody can estimate probability to two and a half points, is the whole business.

The EV calculator runs this for any price and estimate, and the edge calculator reports the gap in percentage points.

EV is arithmetic. Getting a probability worth putting into it is the actual job, and there are only a few honest sources.

• Another market, with the vig removed. If a sharp book prices the same game at −105 and yours is at −110, the fair probability from the sharper market is a defensible estimate. This is the most reliable method available to most bettors and requires no modelling — see the no-vig calculator.
• A model you have tested out of sample. Useful only if it has been checked against results it was not fitted on. A model tuned to past data will report edges that do not exist.
• Information the market has not priced. Real, and rare, and it decays in minutes on liquid markets.

What does not work is confidence. “I think they win this” converted into 60% produces a large EV number that is entirely an artefact of the number you chose. The formula will faithfully compute an edge from a fabricated probability and give you no indication that it did.

Three things to hold onto once you start calculating EV routinely.

Positive EV bets lose constantly. The 55% example loses 45 times in 100. A run of five straight losses in that scenario is unremarkable — over 100 bets it happens about three times in four. Losing streaks are not evidence that the edge was imaginary, which is exactly what makes them dangerous: they feel like evidence.

EV per bet is not EV per dollar of bankroll. A 5% edge tells you the bet is worth making. It does not tell you how much to stake — that is a separate question with its own answer, covered in the Kelly guide.

Small errors dominate. The edge in the example is 2.62 percentage points. If your probability estimate is off by three points — which is a small error by any standard — the bet is negative EV and you will never find out from results. This is why closing line value matters: it checks your estimate against the market’s far faster than your win-loss record can.