Fair odds calculator: fair value from sportsbook prices

Fair odds are the odds a bet would pay if the sportsbook took no cut. To calculate them, convert the posted prices on every side of a market to probabilities, remove the margin so they add up to 100 percent, and convert back. That gives you a reference price for every other number on the screen.

Everything else in value betting depends on that reference. A price can only be "good" compared to something, and the fair price is the something.

Why posted odds aren't fair

Take a moneyline: Team A at -130 and Team B at +110. Convert each to an implied probability:

• -130 means risking 130 to win 100, so the implied probability is 130 / 230 = 56.52 percent.
• +110 means risking 100 to win 110, so the implied probability is 100 / 210 = 47.62 percent.

Those add up to 104.14 percent. Two outcomes can't both be 104 percent likely, and the extra 4.14 points are the book's margin (the vig, or juice). The posted odds describe a market that is a little too expensive on both sides.

The calculation, step by step

1. Implied probability from American odds

• Negative odds: |odds| / (|odds| + 100)
• Positive odds: 100 / (odds + 100)

2. Add them up. The total is the overround. Here it is 1.0414.

3. Remove the margin. The simplest method, multiplicative (also called normalization), divides each probability by the total:

• Team A: 56.52 / 104.14 = 54.27 percent
• Team B: 47.62 / 104.14 = 45.73 percent

4. Convert to fair odds.

• Fair probability above 50 percent: odds = -(p / (1 - p)) x 100. For A: -(54.27 / 45.73) x 100 = -118.7.
• Fair probability below 50 percent: odds = ((1 - p) / p) x 100. For B: (54.27 / 45.73) x 100 = +118.7.

So the fair line for this game is about A -119 / B +119 at one decimal, and the posted -130 / +110 is the book's price with its cut on top.

These are illustrations of the arithmetic, not results.

Which margin-removal method to use

Dividing by the total assumes the book adds the same proportional margin to every outcome. Books don't always do that. Lopsided prices tend to carry more of the margin on one side, and the method you pick changes the answer most on those.

A 2017 study of bookmaker odds compared the additive, multiplicative, Shin and power methods and concluded that the power method universally outperforms the multiplicative method and outperforms or is comparable to the Shin method (American Journal of Sports Science). The power method finds an exponent k so that each implied probability raised to k adds up to 1.

On the example above, k comes out near 1.0625 and the power method gives A at about 54.5 percent and B at about 45.5 percent, a shade different from the multiplicative 54.27 / 45.73. On an even game the methods nearly agree. On a heavy favorite they drift apart.

quant+ removes the margin with the power method by default and offers multiplicative, additive and Shin as alternatives.

Where the fair price should come from

You can devig any one book's market, but the book's own prices include its quirks. A better reference starts from the books that take the most informed money. On the board, the quant+ fair price uses sharp books (Pinnacle and Circa) when they price the market and falls back to a consensus of several books when they don't. If there aren't enough books, it shows no fair price rather than guessing.

Even then, the fair price is an estimate of what the market thinks. It's a strong reference, though: betting odds are frequently found to outperform mathematical models in sports forecasting (PLOS ONE, 2018), and the probabilities implied by public odds are information, enough that researchers have used them to look for mispriced bets (arXiv, 2017). It isn't what will happen. Economist Steven Levitt found that bookmakers set a price and adjust it only occasionally, which is why different books are at different numbers in the first place.

Using fair odds to judge a price

Once you have the fair probability, any price becomes a one-line test. Suppose another book offers Team A at -110.

EV per 100 staked = 0.5427 x 90.91 - 0.4573 x 100 = +3.61

And Team B at +110 at the original book:

EV per 100 staked = 0.4573 x 110 - 0.5427 x 100 = -3.97

The first is priced above the fair line. The second is priced below it. The full walk-through of that formula, with break-even probabilities, is on the expected value calculator page.

Fair odds for parlays and futures

Parlays. For independent legs, multiply the fair probabilities. Two legs at 54.27 percent each give 0.5427 x 0.5427 = 29.45 percent, which is fair odds of about +239.5. The catch is the word "independent". Legs from the same game often move together. If the real chance of both hitting were 33 percent instead of 29.45 percent, the fair price would be about +203, and a payout that looked generous would be ordinary. For same-game parlays, the fair price depends entirely on how well you estimate the correlation, so treat any single number with caution.

Futures. A market with many outcomes, such as a championship winner, carries a margin spread across all of them, and the implied probabilities can add up to well over 100 percent. The method is the same: sum them, then remove the excess. The more outcomes there are, the more a small margin on each adds up.

What the quant+ board does with this

For every game market, the board shows the best price, the book, and the fair value beneath it, along with the edge between them. You don't enter odds into a form to get that. Each row is already devigged. The odds comparison page shows how the board lines up the books, and the positive EV scanner page shows how it ranks prices that beat the fair line.

When you do want to check one by hand, the steps above are the whole job. A calculator saves time. It doesn't change the math.

Limits

• Fair isn't true. The fair price is the market's view with the margin taken out. If the market is wrong about a game, the fair price is wrong with it.
• Stale inputs give stale answers. A fair price built from prices that moved ten minutes ago can mislead.
• Method choice matters most at the edges. Heavy favorites, longshots and multi-outcome futures are where methods disagree.
• Not every market can be devigged cleanly. Alternate lines and thin props may have too few prices to build a reliable fair line.

quant+ shows fair prices for information. It doesn't place bets or connect to sportsbook accounts.

Quant+ is analytics for informational purposes. 21+. If you or someone you know has a gambling problem, call 1-800-GAMBLER.