Expected value per unit, and how big an edge has to be

§01The formula, and what each part is

For a bet that either wins or loses, expected value per unit staked is:

EV = p × (odds − 1) − (1 − p)

Here p is your estimate of the win probability and odds is the decimal price. The first term is what you win when you are right, weighted by how often you are right. The second term is the stake you lose when you are wrong, weighted by how often you are wrong. If the result is positive, the bet is a value bet at those two numbers; if it is negative, it is not.

A worked number makes it concrete. Take a price of 2.20 and an estimate of 50%. Then EV = 0.50 × 1.20 − 0.50 = +0.10. Over one hundred bets of one unit, the arithmetic expects a profit of ten units — in expectation, which is not the same thing as reliably, and not the same thing as soon.

Rearranged, the same formula answers the other question: at this price, what chance do I need to believe before the bet is worth making? That is the break-even rate, and it is simply 1 / odds, the implied probability from the previous page.

§02Break-even rate — the only comparison that matters

The implied chance of a price is the rate at which it breaks even. Price 2.20 breaks even at 45.5%, so a bettor who believes the true chance is 50% has an edge of 4.5 percentage points and an expected value of +10% of stake. Price 1.50 breaks even at 66.7%; to have the same +10% edge you would need to believe the chance is 73.3%, which is a much larger claim about the world.

This is why comparing edges across prices has to be done in percentage points against the break-even rate, not in raw odds movement. A 4.5-point edge at a 2.20 price and a 4.5-point edge at a 6.00 price are the same size of claim about the world; they are not the same size of payout, and they are certainly not the same reliability.

Two things to notice. The edge in points and the edge in expected value are different measurements, and either can be made to look impressive. And a smaller point edge at a longer price produces the same expected value from a much shakier estimate, because long prices attract wide margins and thin information.

§03Edge size, sample size, and how long the wait is

A positive expected value tells you the direction of a long-run average. It says nothing about the next fifty bets. The standard deviation of a series of even-money bets is roughly one unit per bet, which is large compared with a per-bet edge of 0.10 units. The result is that a genuinely profitable bettor spends a great deal of the time behind.

There is an approximate rule of thumb worth carrying: the number of bets needed before your results are a reasonable reflection of your edge grows roughly with the inverse square of the edge. A 10% edge becomes legible in tens of bets, a 2% edge in thousands. Most bettors who measure a short sample and conclude they have no edge are measuring noise and drawing a conclusion about signal.

§04One bet, written out

A market prices a selection at 2.40, which implies 41.7%. After removing the amount taken by the operator — see the previous page for the method — the de-margined market chance is 43.5%. You believe, from whatever work you consider legitimate, that the true chance is 47%. The implied break-even rate is 41.7%, so the market has already built in a 1.8-point cushion over its own fair estimate, and your own edge over the price is 5.3 points.

The last line is the one people misread. It is an average over a hypothetical very long run, not a forecast. Whether any individual bet wins is decided by the event, not by the arithmetic.

§05Where stake size enters, and where this site stops

Edge determines whether a bet has positive expected value; stake determines whether you survive the variance long enough to collect it. Staking more than a small fraction of a bankroll on a positive-EV bet can still be a losing proposition in practice, because a run of losses at a large stake is unrecoverable. The Kelly criterion, which sizes stakes in proportion to edge divided by odds, is the standard reference; most practitioners deliberately stake well below the full Kelly amount, precisely because the edge estimate is uncertain.

This site deliberately does not give staking instructions. It gives the reason staking matters: once you know the edge is positive, the only remaining ways to lose are to be wrong about the edge, or to bet an amount that variance can destroy. Both are real.