American odds to implied probability (and what −110 actually means)

Every price is a probability wearing a costume. −110, 1.91, 52c and 52.4% are four ways of writing the same thing, and the only reason there are four is history — American books, European books and prediction markets each grew up separately. Converting between them is two lines of arithmetic, and doing it is the difference between having an opinion and being able to price one.

What −110 means

A minus number is what you must risk to win $100. −110 means risking $110 to win $100. A plus number is what you win from a $100 stake: +150 means risking $100 to win $150. That is the whole notation.

So −110 is not "a 110% chance" of anything, and it is not the bookmaker's fee. It is a payout ratio, and the probability it implies is 52.38%.

The two formulas

For a negative price (the favourite), divide the number by itself plus 100:

implied % = (−odds) ÷ ((−odds) + 100) → 110 ÷ 210 = 52.38%

For a positive price (the underdog), divide 100 by the number plus 100:

implied % = 100 ÷ (odds + 100) → 100 ÷ 250 = 40.0%

Decimal odds are easier still: the implied probability is simply 1 ÷ decimal. And a prediction market price needs no conversion at all — 52c *is* 52%, which is the single best reason to learn to read one.

Conversion table

The adjustment almost everyone forgets

Convert both sides of a −110 / −110 market and you get 52.38% and 52.38%. That sums to 104.76%, and probabilities cannot sum to more than 100%. The extra 4.76 points is the bookmaker's margin, baked into both prices.

That correction is worked through properly in what the vig really costs you, including what it does to the hit rate you need just to break even.

Break-even, by price

The implied probability *is* your break-even rate. Hit it exactly and you finish level; the gap between it and your real hit rate is your edge.

This is why "I hit 55% of my bets" is not, on its own, a claim about anything. At −110 that is a real edge. At −150 it is a losing season.

Doing it without the arithmetic

Sharply's odds comparison converts and de-vigs every book's price on a game at once, so you can see the fair number and which book is furthest from it without touching a calculator. And when you run a market through the analyser, the de-vigged price is what its estimate is measured against — never the raw one.