Expected value, from scratch

The only number that separates a good bet from a lucky one — and the one calculation that has to be right before any other tool is worth using.

Expected value is what a bet is worth on average, if you could make it a thousand times. It is the only number that separates a good bet from a lucky one, and it is the calculation every other tool in this section is built on.

A bet can be good and lose. A bet can be terrible and win. Expected value describes the bet; the result describes one evening.

The formula

For a $1 stake: EV = (your probability × profit if it wins) − (chance it loses × 1).

A worked example

A book offers +120 on a game.
You believe the true chance is 50%.
+120 pays $1.20 profit per $1 risked.
EV = (0.50 × 1.20) − (0.50 × 1) = 0.60 − 0.50.
= +0.10 per dollar.

Ten cents of expected profit on every dollar — a 10% edge on stake. On $100, that is +$10 in the long run.

There is a second, quicker way to see the same thing. +120 implies 45.45%. You think it is 50%. The 4.55-point gap between those is your edge, and everything else follows from it.

The mistake that makes every EV calculator lie

Your probability cannot come from the price you are judging

If you take a book's price, convert it to an implied probability, and feed that back in as your "true" number, you will find a small negative edge on every bet you ever check — because you have compared the price against itself plus the fee. The fair number has to come from somewhere else.

In practice "somewhere else" means one of three things:

A sharp book's price, de-vigged. The standard approach, and what most tools do.
A consensus across several de-vigged books. Steadier than any one of them.
Your own model. The hardest, and the only one that can find an edge a de-vig cannot.

How big an edge is realistic

What different edges mean in practice
Edge	What it usually is	How to treat it
Under 1%	Noise, or the two books rounding differently	Ignore
1-3%	A genuine soft price, or a slightly stale one	The realistic working range
3-6%	A stale line, or a real disagreement	Worth taking, worth double-checking
Over 8%	Usually an error — wrong player, wrong period, wrong market	Check before you bet, not after

The instinct is to get excited about the largest number on a screen. The correct instinct is the opposite: a 15% edge on a major market almost always means the two prices are not describing the same bet.

Why it feels like nothing is happening

A 2% edge is real and it is also slow. Bet a hundred times at 2% and you expect two units of profit — against a swing of plus or minus ten units from variance alone. The edge is invisible inside the noise for a long time.

This is not a reason to bet bigger. It is the reason closing line value exists as a measurement, and the reason the next lesson is about it.

The same calculation, run across every book and every market on the slate. Open the +EV scanner.