Implied Probability in Betting: Formulas, Margin and Fair Odds

Implied probability is the chance of an outcome that a price represents. More precisely, it is the win rate at which a bet at that price would break even over the long run. A price of 2.75 implies 36.36%: win more often than that at 2.75 and you profit over time, win less often and you lose.

That number is the bridge between odds and judgement. You cannot easily compare 7/4 with a feeling that a team has "a decent chance", but you can compare 36.4% with your own estimate of 40%. Our guide to how betting odds work introduces the basic formulas. This article goes further: converting probabilities back into odds, break-even rates on standard prices, exchange commission, overround versus hold, and three ways of removing the bookmaker's margin to find a fair price.

Implied probability formulas for every odds format

All three formats lead to the same number. Take one price written three ways: 7/4, 2.75 and +175.

• Fractional a/b: b ÷ (a + b). For 7/4, 4 ÷ 11 = 36.36%.
• Decimal: 1 ÷ odds. For 2.75, 1 ÷ 2.75 = 36.36%.
• American plus: 100 ÷ (odds + 100). For +175, 100 ÷ 275 = 36.36%.
• American minus: odds ÷ (odds + 100), ignoring the sign. For -130, 130 ÷ 230 = 56.52%, the same as 10/13 in fractions or 1.77 in decimal.

Decimal is the quickest format to work with, so if you bet in fractions or American odds it is worth switching the display in your account settings, or at least converting to decimal before you do any sums.

Turning a probability back into odds

The reverse step is just as useful, because it tells you the price you need before you look at the market. Suppose you rate a team at 62%.

The fair decimal price is 1 ÷ 0.62 = 1.613. As a fraction that is (1 - 0.62) ÷ 0.62 = 0.613 to 1, close to 8/13, which is 1.615 in decimal. In American odds, for probabilities above 50% the formula is -(p ÷ (1 - p)) × 100, which gives -(0.62 ÷ 0.38) × 100 = -163. For a 30% chance the fair decimal price is 3.33, the fraction is 7/3 and the American price, using ((1 - p) ÷ p) × 100, is +233.

These are fair prices for your estimate. Anything shorter is a bet you expect to lose money on, if your estimate is right. Anything longer is a candidate, which is where value betting starts.

Break-even win rates on common prices

Implied probability is also the break-even rate you need over a run of similar bets. This matters most in markets where you see the same price again and again, such as spreads and totals:

• -105 (1.95): 51.22%
• -110 (1.91): 52.38%
• -115 (1.87): 53.49%
• -120 (1.83): 54.55%

The gap between -105 and -120 is more than three percentage points of win rate. That is a lot when a skilled bettor's edge is often only a few points. The same arithmetic explains why taking the best available price matters: a bet at 2.40 needs a 41.67% chance to break even, the same bet at 2.30 needs 43.48%.

Betting exchanges add one more step, because they charge commission on net winnings instead of building a margin into the price. Backing at 2.20 with a 2% commission (rates vary by exchange and market) pays 1.20 × 0.98 = 1.176 profit per unit, an effective price of 2.176. The break-even rate rises from 45.45% to 45.96%. Always work from the effective price, not the one on the screen.

Overround and hold: two ways to measure the margin

Add up the implied probabilities of every outcome in a market and you get more than 100%. The excess is the overround. A tennis match at -110 on both players totals 52.38% + 52.38% = 104.76%, an overround of 4.76%.

A second figure, the hold, answers a slightly different question: what share of all money staked the bookmaker expects to keep if bets come in proportion to the prices. Hold = 1 - (1 ÷ total implied probability). For the -110 market that is 1 - (1 ÷ 1.0476) = 4.55%. Hold is the more common term in the US, while UK bettors tend to talk about overround. Both describe the same margin from different angles, and hold is always the smaller number.

The margin grows with the number of outcomes. Here is a six-runner market of the kind you might see in a small horse race or a novelty market:

• 3.00: 33.33%
• 4.00: 25.00%
• 5.00: 20.00%
• 6.00: 16.67%
• 8.00: 12.50%
• 10.00: 10.00%

The total is 117.5%, an overround of 17.5% and a hold of 14.89%, against 4.76% on the two-way tennis match. Outright and correct score markets with dozens of outcomes often carry more still, which is why a long price in those markets can look generous and still be poor value.

Removing the margin: three no-vig methods

To estimate the probabilities the market actually believes, you strip the margin out. The result is called the fair, no-vig or vig-free price. There is no single correct method, because nobody outside the bookmaker knows exactly how its margin was spread. Three common approaches give slightly different answers.

Take a football (soccer) match priced Home 1.62, Draw 4.20, Away 5.50. The implied probabilities are 61.73%, 23.81% and 18.18%, a total of 103.72%.

The proportional method divides each figure by the total: 59.51%, 22.96% and 17.53%, or fair odds of 1.68, 4.36 and 5.70. It assumes the margin is spread in proportion to each outcome's chance.

The additive method subtracts an equal share of the 3.72-point excess from each outcome, 1.24 points each: 60.49%, 22.57% and 16.94%, or fair odds of 1.65, 4.43 and 5.90. It puts relatively more of the margin on the long shot, which fits the favourite-longshot bias, but in markets with many outsiders it can push a long shot's probability below zero, so it only suits short markets.

The power method raises every implied probability to the same power k, chosen so the results add up to exactly 100%. Here k is about 1.04, giving 60.54%, 22.48% and 16.98%, or fair odds of 1.65, 4.45 and 5.89. It also shifts margin towards the outsiders and never produces negative numbers, which is why many bettors prefer it for bigger fields.

The differences look small, but they decide bets. Under the proportional method, an away price of 5.80 at another bookmaker looks like value against a fair 5.70. Under the other two methods it does not. More advanced approaches exist, such as the model associated with the economist Hyun Song Shin, which treats part of the margin as the bookmaker's protection against better-informed bettors, but the three above cover most practical needs.

Fair prices in a two-way market

Two-way markets are simpler, and the proportional method is the usual choice. A baseball moneyline at -165 and +140 implies 62.26% and 41.67%, a total of 103.93%. Dividing through gives fair probabilities of 59.91% and 40.09%, which convert to fair prices of about -149 and +149, or 1.67 and 2.49 in decimal. In other words, the favourite asks you to risk 165 to win 100 where the fair figure is about 149, and the underdog pays 140 where the fair figure is about 149. Bettors who track closing line value run exactly this calculation on the closing market.

Implied probability of parlays and combined bets

For a parlay or accumulator, multiply the decimal prices and take 1 ÷ the result. Three legs at 1.91 multiply to 6.968, an implied probability of 14.35%. If each leg is really a 50% chance, the true probability is 0.5 × 0.5 × 0.5 = 12.5%, a fair price of 8.00. The parlay pays 6.97, about 87% of the fair price, because the margin on each leg compounds. The parlay and accumulator guide covers this in more depth.

Same game parlays and bet builders work differently. Their legs are often linked, for example a team winning and its main striker scoring, so the true probability is not the simple product of the legs. Bookmakers price that correlation in themselves, usually with a bigger margin than on single markets, which makes the implied probability of the combined price much harder to check at home.

Where beginners go wrong with implied probability

The most common error is reading implied probability as the true chance. It is the bookmaker's price including margin, so it overstates the chance of every outcome in the market. A team at 1.62 does not have a 61.73% chance in the market's view; with the margin removed it is nearer 59.5% to 60.5%, depending on the method.

The second is comparing numbers across markets with different margins. A 20% implied chance in a two-way market and a 20% implied chance in a 30-runner outright are not the same reading, because the outright carries far more margin. The third is rounding too early. Working from 1.91 instead of 1.909, or 52% instead of 52.38%, is fine for one bet but distorts a record of hundreds. The fourth is the minus sign on American odds: -130 is a favourite at 56.52%, not a negative number to add or subtract.

Before you place the bet: a worked example

Here is the full routine on a single bet. A football match is priced Home 2.40, Draw 3.30, Away 3.10, and you rate the home side at 45%.

• Implied probabilities: 41.67%, 30.30% and 32.26%. The total is 104.23%, so the overround is 4.23%.
• Fair proportional probabilities: 39.98%, 29.07% and 30.95%.
• Your break-even rate at 2.40 is 41.67%. Your estimate of 45% clears it by 3.3 points.
• Expected value: (0.45 × 2.40) - 1 = +8% per unit staked.
• Sanity check: the market's fair view is about 40%, five points below yours. Write down what you know that the market does not. If you cannot, lower your estimate before you bet.

If the bet still stands, size it with a fixed staking plan from our bankroll management guide and check that the operator is licensed where you live, since laws differ by country and, in the US and India, by state, and some Indian states ban online betting. Implied probability tells you what a price demands; it does not make any result more certain. Our responsible gambling page has limit tools and support if you need them.