Expected value in betting: what +EV means & why most bets lack it
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How expected value is calculated for a single bet, how the bookmaker’s margin pulls it below zero, & why even a positive figure only shows up over many bets.
What expected value means
In probability theory, Wikipedia’s entry describes expected value (“also called expectation, mean, or first moment”) as “a generalization of the weighted average”. Where there are a limited number of outcomes, it is the average of those outcomes with each one weighted by its probability.
Wikipedia adds that the expected value “can be interpreted as the long-run average of results from independent repetitions of the same random experiment, as formalized by the law of large numbers”. The law of large numbers is the result that, as an experiment is repeated more & more times, the average of the results settles towards the expected value.
In betting shorthand, “+EV” means positive expected value: a bet whose average result over many repeats would be a profit. “-EV” means the average result would be a loss.
The idea is old. According to Wikipedia, it grew out of Blaise Pascal’s & Pierre de Fermat’s work on a gambling puzzle in 1654, built on the principle that “the value of a future gain should be directly proportional to the chance of getting it”.
The formula for a single bet
A bet with two outcomes, win or lose, has a simple expected value. You multiply the chance of winning by the profit if it wins. Then you take away the chance of losing multiplied by the stake.
Written out, the formula is: EV = (probability of winning x profit) – (probability of losing x stake).
Wikipedia uses American roulette as an example. The wheel has 38 pockets. A $1 bet on a single number pays $35 if it wins, “which happens with probability 1/38”, & loses the $1 otherwise.
Wikipedia works out the expected profit from those figures: “the expected value to be won from a $1 bet is −$1/19”, a loss of a little over five cents per dollar on average.
A worked example
Take a hypothetical £10 bet that would return £11 profit if it wins. In the decimal odds used in Karl Whelan’s paper on parlays, a stake s at odds D wins (D – 1)s, so this is a price of 2.10.
To find the expected value, multiply the estimated probability of the bet winning by the £11 profit. Then subtract the estimated probability of losing multiplied by the £10 stake.
Now suppose the same outcome is offered at a lower price that pays less profit. The first part of the sum gets smaller while the second stays the same, so the expected value falls. A low enough price turns it negative.
The event & the probability have not changed. Only the price has, & that alone can move a bet from positive to negative expected value. The whole calculation also depends on the probability estimate being right.
Fair odds: taking out the margin
The probability that goes into the formula has to be a fair one: an estimate of the real chance, not the chance implied by the bookmaker’s price. Lisandro Kaunitz, Shenjun Zhong & Javier Kreiner explain in their paper that “for the expected value of a bet to be zero, the odds paid by the bookmaker must be the inverse of the underlying probability of the result.”
They use a roulette bet on red, which wins 18 times in 38. The fair decimal price would be 2.111, “but the house pays only 2”. They call the gap “the ‘tax’ or commission charged by the bookmaker”, known as the margin.
The Wizard of Odds sets out a method for measuring that margin across a market. First, convert each price to a probability: “if the odds pay x to y then the probability of winning is y/(x+y)”, so 35 to 1 becomes 1/36. Then add all the probabilities to get a total, t. The house edge is 1-(1/t).
Its example is the odds the Suncoast casino in Las Vegas offered on the Pontiac Excitement 400, run on 5 May 2001. Dale Jarrett at 4 to 1 converts to 20.00%, & the whole field adds up to a “total probability of 135.04%”.
A fair book would add up to 100%. The Wizard of Odds works out the house edge as “1-1/1.3504=0.259496714, or 25.95%”. This means the probabilities implied by the prices overstate the real chances, & the margin has to be removed before they can stand in as fair probabilities.
Why most bets carry negative EV
Whelan, of University College Dublin, writes: “Sportsbooks build in a profit margin on their odds, so on average they come out ahead.” If the price is set below fair value, a bettor whose probability matches the bookmaker’s fair estimate has a negative expected value.
Combining bets multiplies the effect. Whelan notes that “winning a parlay requires you to beat this margin multiple times”. A parlay is a multi-leg bet that pays only if every leg wins. He reports that state gaming commissions find “the fraction of stakes lost for bettors is about four times higher for parlays than other bets”.
His table for New Jersey shows sportsbook revenue as a share of the money staked. From 2021 to 2025 it ranged from 17.5% to 19.2% on parlays, against 3.3% to 5.0% on other bets.
The Wizard of Odds puts the general point in its Ten Commandments of Gambling: “Thou shalt expect to lose.” It adds: “Even with good rules and strategy the odds are still usually in the casino’s favor.”
What a positive-EV bet looks like
A positive-EV bet is one where the price is above fair value: the odds pay more than the true probability justifies. Spotting one means having a fair probability that is more accurate than the price offered.
Kaunitz, Zhong & Kreiner did not build their own forecasting model. Instead, they used bookmakers’ “publicly available odds as a proxy of the true probability of a game outcome”. They then searched for “games with odds offered above the estimated fair value”.
To build that proxy, they averaged closing odds across up to 32 bookmakers. The data covered 479,440 football games from 818 leagues & divisions between January 2005 & June 2015.
The authors report that their strategy “proved profitable in a 10-year historical simulation using closing odds”. It also made a profit in a 6-month simulation using minute-by-minute odds & over 5 months of betting real money.
That did not last. “A few weeks after we started trading with actual money some bookmakers began to severely limit our accounts, forcing us to stop our betting strategy,” they write.
Variance & sample size
Expected value describes an average, not a single result. A single bet either wins or loses, & the gap between expected & actual results narrows only over many bets.
Even Wikipedia’s roulette example is worded with care: “in 190 bets, the net loss will probably be about $10.” The word “probably” matters. Over that many spins, a player could finish well above or below the expected loss.
Variance also cuts the other way. In a study of betting on a biased coin, 61 “young, quantitatively trained men and women” were each given $25 & 30 minutes to bet on a coin they were told was “biased to come up heads 60% of the time”.
The authors describe it as a game with “favourable odds”. Even so, “About 30% of the subjects actually went bust, losing their full $25 stake,” & the majority “did not place their bets very well”.
EV, stake size & limits
The coin study suggests that a favourable edge does not by itself protect a bettor from losing everything. How much is staked on each bet also matters.
Whelan’s paper sets out the Kelly criterion, a rule for sizing stakes. Under it, a bettor stakes the fraction (pD – 1)/(D – 1) of their wealth, where p is their probability & D the decimal odds. He notes that the stake “increases with the size of the perceived edge pD − 1 and decreases with the odds”.
Whelan also cites Arrow (1971, page 100), who shows that a risk-averse person will accept a positive expected value gamble “once the stake is sufficiently small”. Both points tie the right stake to the size of the edge & the risk of the bet.
The Wizard of Odds’ commandments include “Thou shalt not over-bet thy bankroll”. It advises: “Stick to your limits and don’t gamble with money you need for necessities.” On losses, it says: “Think of it as the price you pay for entertainment.”
Questions
What does +EV mean in betting?
It means positive expected value: a bet whose probability-weighted average result is a profit. Wikipedia describes expected value as the long-run average of results from repeating the same random experiment.
How do you calculate the expected value of a bet?
Multiply the probability of winning by the profit, then subtract the probability of losing multiplied by the stake. Wikipedia’s example of a $1 single-number bet in American roulette, paying $35 with probability 1/38, gives an expected value of -$1/19.
How does the bookmaker’s margin affect expected value?
The margin means prices are set below fair value. The Wizard of Odds shows prices for one 2001 race implying a total probability of 135.04%, a house edge of 25.95%.
Can a positive-EV bettor still lose money?
Yes. In one study of a coin biased to land heads 60% of the time, about 30% of the 61 players lost their full $25 stake.
Do bookmakers restrict bettors who win?
Kaunitz, Zhong & Kreiner report that a few weeks after they began betting real money with a strategy targeting prices above fair value, some bookmakers began to severely limit their accounts.
Sources
Sources checked 30 October 2026. Information only. 18+.
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