Positive EV Betting: How to Calculate Your Edge

By · author of the MIT-licensed teachersbettextbook library ·

Two hamsters. Dead even race. So why does the board never pay like it?

Meet your classmates

Same five in every lesson. Only the situation changes.

Today Brian is taking bets on a race between Chalk and Pip. He is not trying to guess the winner. He wants money on both hamsters, so that whoever wins, he is fine and he keeps a slice.

So he opens both at −110

−110 means you risk $1.10 to win $1.00. On a coin flip that is a bad deal, and it is a bad deal on both hamsters. That slice is called the vig. It is how he eats.

Then your whole class piles on Pip

Now he is holding a stack of Pip tickets and almost nothing on Chalk. If Pip wins he is wiped out. He is not worried Pip is better. He is worried he is lopsided.

So he moves the price:

The hamsters never found out. Still a coin flip. But Chalk now pays more than a coin flip is worth, and that is the whole game.

Watch what happens

Pip and Chalk run all afternoon. Joe never waits for anything, so he is in at the opening −110 before Brian moves a thing. He is part of the reason it moves. Evan is the one still standing there when it does, and he takes Chalk at +110. Same race, same two hamsters, ten seconds apart, and one of those tickets is worth more than it cost.

They both start with $10 and bet a dollar a race.

Average Joe Bookie Brian +EV Evan Joe, on Pip at −110 Evan, on Chalk at +110 $10 $0 $0.00 $21.90 49 races later, Joe is out of lunch money. $0.00 $21.50 41 races later, Joe is out of lunch money. $0.00 $21.40 39 races later, Joe is out of lunch money. $0.00 $20.60 44 races later, Joe is out of lunch money. $0.00 $21.40 39 races later, Joe is out of lunch money. $12.75 $6.00 45 races, and this time it is Evan who is down. Better price, worse day. That happens too. Everyone likes Pip. Joe is everyone. Brian moves the price. Evan takes the other side. Joe goes home hungry.
A real simulation, not a drawing, and one afternoon out of six picked at random. Both charts are on the same scale and both men started on the dashed $10 line. Joe took Pip at the opening −110, which on a coin flip loses 4.5¢ per dollar. Evan waited for Chalk at +110 and made 5¢. Half a cent a race is nothing, right up until the bell goes.

Some afternoons Chalk wins more, some afternoons Pip does. Over enough races it evens out. Both hamsters really are 50/50.

What never evens out is the price. Every single race, Joe was paying 4.5¢ on the dollar for the privilege of being in, and Evan was being paid 5¢. That is under a dime a race. It sounds like nothing. It does not care who wins, it just grinds, and by home time it is somebody's whole lunch.

You are not trying to guess which hamster is faster. You are watching for the moment the board stops matching the race.

Want more math?the Nerd Corner

Nerd Corner

Everything above is one equation. It is worth seeing it rearranged, because each form answers a different question and you will meet all of them.

From decimal odds

Decimal odds dd include your returned stake, so d=b+1d = b + 1 and the loss term folds away:

EV=p d−1EV = p\,d - 1

As an edge over the fair price

If qq is the probability the price implies, the expected value is just how far your honest number sits above it, scaled by the payout:

EV=(p−q) dEV = (p - q)\,d
This is why a 2-point disagreement matters more at long odds than at short ones: the same gap gets multiplied by a bigger dd.

Break-even

Set expected value to zero and solve for the probability that makes the price fair:

p∗=1d=1b+1p^{*} = \frac{1}{d} = \frac{1}{b + 1}

At −110 that is 1/1.909=52.4%1 / 1.909 = 52.4\%. You need to be right more than 52.4% of the time just to break even, on a race that is 50/50. That gap is the cut.

When the number is too good

An edge above about 15 cents on the dollar is almost never an edge. It is a stale price, a typo, or a market that has already moved and not been taken down. This site refuses to flag those rather than dressing them up as opportunities.

Check your understanding

Frequently asked questions

What is a +EV bet?

The offered price beats the outcome's fair probability, so the average result per dollar staked is positive.

How do you calculate expected value in betting?

EV = p · b − (1 − p): p is the fair win probability, b the decimal profit per dollar. Positive means the price favors you.

How do you find the true probability for an EV calculation?

De-vig a sharp book's line for a fair benchmark, then compare a softer book's implied probability against it.

Why do +EV opportunities disappear so fast?

Typical 1%–4% edges are the same size as normal line movement, so a +EV price decays within minutes.