Decimal odds become easier to understand when we start with an actual stake. This makes the formula implied probability = (1 ÷ decimal odds) × 100% intuitive, but one distinction matters: the percentage derived from the odds is not a certain measure of the event’s true probability. We will use the Arsenal–Tottenham 1–X–2 odds in an information note from Italy’s Customs and Monopolies Agency (ADM).
Odds of 1.9: what does each side risk?
If you stake €1 at odds of 1.9 and lose, you lose €1. If you win, you receive a €1.90 gross return, including your original euro: you gain €0.90 net and, considering this bet in isolation, the bookmaker loses €0.90 net. We need not imagine an advance payment into a common pot: we compare what each party could lose.
Dividing the amounts at risk gives the percentage
The possible losses are €1 for you and €0.90 for the bookmaker. Together they make 1+0.90=€1.90. Your share of this risk comparison is [1÷(1+0.90)] × 100% = (1÷1.90) × 100% = 52.63%; the bookmaker’s is [0.90÷(1+0.90)] × 100% = 47.37%. The 52.63% is the bettor’s break-even implied probability; the 47.37% is its complement in this comparison, not the bookmaker’s margin.
Whole-number odds make the rule especially clear. At odds of 2, you risk €1 and the bookmaker risks €1 net: [1÷(1+1)] × 100% = 50%. At odds of 4, you risk €1 and the bookmaker risks €3 net: [1÷(1+3)] × 100% = 25%. In general, for a €1 stake and decimal odds q, the other possible loss is q−1, so your fraction is 1÷[1+(q−1)]=1÷q and the percentage is (1÷q) × 100%.
We can also check this using the break-even point over many identical bets. If p is the chance of winning, p×0.90=(1−p)×1 gives p=1÷1.90. This is a threshold derived from the odds, not a certain measure of the event’s true probability: a different actual chance changes the average outcome.
Apply the method to all three outcomes
In the official example, the odds are 1.9 for an Arsenal win (1), 3.4 for a draw (X) and 4 for a Tottenham win (2). For each outcome, start with €1: the gross return equals the decimal odds and the net profit if it wins is odds−1.
| Outcome | Odds | Net profit if it wins (on €1) | Implied percentage |
|---|---|---|---|
| 1 | 1.9 | €0.90 | (1÷1.9) × 100% = 52.63% |
| X | 3.4 | €2.40 | (1÷3.4) × 100% = 29.41% |
| 2 | 4 | €3.00 | (1÷4) × 100% = 25.00% |
The same equation applies to every row: p × (odds−1)=(1−p) × 1, hence p=1÷odds. To express it as a percentage, show the step (1÷odds) × 100%.
The book exceeds 100%
Add the unrounded values: 52.631578…% + 29.411764…% + 25% = 107.043343…%. This sum is the book percentage or overround; the excess over 100% is 7.043343… percentage points. Do not confuse that with a loss of 7.04% of all money staked: the denominators differ.
If we choose to remove the margin proportionally, divide each implied percentage by 1.07043343… (equivalently, divide by 107.043343… and multiply by 100). We get 49.17% for 1, 27.48% for X and 23.36% for 2, rounding only at the end; the three rounded figures therefore add up to 100.01% rather than 100%. These are odds-based normalized estimates, not true probabilities: the odds alone do not reveal how the bookmaker allocated its margin or the outcomes’ actual chances. The ADM note’s table gives 49.1% for 1 because it uses already-rounded intermediate values; the exact figures give 49.168…%.
Why is the theoretical payout 93.42%?
Imagine distributing €100 in total across the three outcomes so that the gross return is the same whichever one occurs. Stakes are proportional to 1÷odds: about €49.17 on 1, €27.48 on X and €23.36 on 2 (rounded cents may not sum to exactly €100). Before rounding the stakes, the gross return is always 100÷1.07043343… = €93.42. The difference is €6.58, or 6.58% of the €100 staked. This is the market’s theoretical payout at those odds, not a promise about the return on an individual bet.
Try two different odds
Two mutually exclusive and exhaustive outcomes have odds of 1.6 and 2.5. Find their implied percentages, the book percentage and the theoretical payout if you cover both outcomes proportionally.
Worked answer
The implied percentages are (1÷1.6) × 100% = 62.5% and (1÷2.5) × 100% = 40%. The book is 102.5% and its excess is 2.5 points. The theoretical payout is (1÷1.025) × 100% = 97.56%. Normalizing the two percentages gives 60.98% and 39.02%: again, these are estimates derived from odds, not known true probabilities.
Remember: decimal odds multiply the stake to give the gross return; the break-even threshold is 1÷odds; adding the thresholds for all mutually exclusive and exhaustive outcomes gives the book percentage. Studying these ratios is an arithmetic exercise, not a method that guarantees a win.