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From a price to a probability, and where the bookmaker's margin sits

Three football prices turned into implied chances, the overround they add up to, and an estimate of the prices with the margin taken out.

A three-way football market: home 2.10, draw 3.40, away 3.60.

A football match has three possible results at full time: a home win, a draw or an away win. In this example the three prices are 2.10 for the home win, 3.40 for the draw and 3.60 for the away win, which as fractions are 11/10, 12/5 and 13/5. Every figure that follows is worked out from these three prices alone.

Home win: 1 ÷ 2.10 = 47.62%, the chance the price implies.

A decimal price includes the stake, so it converts straight into a probability: divide 1 by the price. For the home win, 1 ÷ 2.10 = 0.4762, or 47.62%. This is the implied probability, the chance of a home win that the price corresponds to. It is not a forecast of the result, and it still contains the bookmaker's margin.

Draw: 1 ÷ 3.40 = 29.41%. Away: 1 ÷ 3.60 = 27.78%.

The same division works for the other two results. The draw at 3.40 implies 1 ÷ 3.40 = 29.41%, and the away win at 3.60 implies 1 ÷ 3.60 = 27.78%. The longer the price, the smaller the implied chance: a price of 3.60 says the away win is less likely than the home win at 2.10.

Added together they make 104.81%: 4.81 points more than 100%, and that excess is the margin.

Exactly one of the three results will happen, so true probabilities for them would add up to 100%. The implied ones do not: 47.62% + 29.41% + 27.78% = 104.81%. The 4.81 percentage points above 100% are the overround, also called the bookmaker's margin. It is built into all three prices at once, which is why one price on its own cannot show it.

Divide each chance by 1.0481 and all three shrink in proportion: 45.43%, 28.06%, 26.50%.

One way to estimate the chances without the margin is proportional removal: divide each implied chance by the total, 104.81% or 1.0481. The home win becomes 45.43%, the draw 28.06% and the away win 26.50%. Each result gives up the same share of its own chance, so the home win loses 2.18 points and the away win 1.27. This is an estimate, not the bookmaker's own figure.

Add the fair chances back up: 100.00% before rounding, 99.99% as printed.

A check on the arithmetic: the three estimated fair chances describe a complete market, so they must add up to 100%. Before rounding they add up to exactly 100.00%, and the stacked bar now ends on the 100% line. The printed figures, each rounded to two decimals, add up to 99.99%; the missing 0.01 is rounding, not margin left behind.

Back into prices: about 2.20, 3.56 and 3.77, against 2.10, 3.40 and 3.60 quoted.

Dividing 1 by each estimated fair chance gives an estimated fair price: 1 ÷ 45.43% is about 2.20 for the home win, 3.56 for the draw and 3.77 for the away win. Every quoted price is a little shorter than its estimate, on all three results, because the margin applies whichever result is backed. Other methods share the margin out unevenly and give slightly different estimates.

Two outcomes at 10/11 each: 52.38% + 52.38% = 104.76%.

The same arithmetic works in any complete market. Two-way markets are often priced 10/11 on both sides, which is 1.91 in decimal. For a fraction, the implied chance is 11 ÷ (10 + 11) = 52.38%. Both sides together make 52.38% + 52.38% = 104.76%, an overround of 4.76 points. Spread proportionally, the margin leaves each side at an estimated fair 50.00%.

A fair coin at evens both ways: 50.00% + 50.00% = 100.00%, no overround.

Take the margin away and see what changes. A coin toss priced fairly would be evens on each side, 2.00 in decimal, so each side implies 1 ÷ (1 + 1) = 50.00%. The two add up to exactly 100.00% and the overround is 0.00 points. Real markets from a bookmaker add up to more than 100%, and the excess is how the bookmaker is paid for offering them.

Sources: Odds converter (decimal, fractional and implied probability, the same arithmetic as every figure here) · Betting odds explained: the overround and estimating fair prices · Arithmetic: implied chance = 1 ÷ decimal price; estimated fair chance = implied chance ÷ the sum of all implied chances (proportional removal, one of several methods)