In betting, the odds are the price of an outcome: they show how much the organiser pays for a stake if the outcome happens. The odds also reveal the probabilities the organiser assigns to the outcomes, and the margin it keeps. This page explains the three usual formats of odds, the conversions between them and the mathematics of the margin. The odds used as an example are invented for this page and do not come from any organiser.
Decimal odds
Decimal odds show the amount returned for each unit staked, stake included. At odds of 2.10, a stake of 10 returns 21 if the bet wins: the 10 of the stake and 11 of net winnings. If the bet loses, the whole stake is lost.
- total return = stake × odds
- net winnings = stake × (odds − 1)
Decimal odds are always greater than 1. Odds of 2.00 double the stake; odds of 1.50 bring net winnings equal to half the stake. The decimal format is the usual one in Romania and in most of continental Europe.
Fractional odds
The fractional format, traditional in the United Kingdom and Ireland, shows only the net winnings relative to the stake. Odds of 11/10 mean 11 units of net winnings for 10 units staked; 5/2 means 5 for 2; 1/2 means 1 for 2, that is, net winnings smaller than the stake. The stake is returned on top. The conversion is direct:
- fractional to decimal: decimal odds = fraction + 1 (11/10 becomes 2.10; 5/2 becomes 3.50; 1/2 becomes 1.50);
- decimal to fractional: fraction = decimal odds − 1 (2.00 becomes 1/1, known as "evens"; 3.00 becomes 2/1).
American odds (moneyline)
The American format uses signed whole numbers, relative to an amount of 100:
- positive odds, for example +150, show the net winnings on a stake of 100; the decimal odds are 1 + 150/100 = 2.50;
- negative odds, for example −125, show how much must be staked for net winnings of 100; the decimal odds are 1 + 100/125 = 1.80.
The other way round, decimal odds of 2.00 or more are written in American format as +100 × (odds − 1), so 2.10 becomes +110. Decimal odds below 2.00 are written as −100 / (odds − 1), so 1.50 becomes −200. Odds of 2.00 correspond to +100.
The same prices in the three formats:
| Decimal | Fractional | American |
|---|---|---|
| 1.50 | 1/2 | −200 |
| 1.80 | 4/5 | −125 |
| 2.00 | 1/1 | +100 |
| 2.10 | 11/10 | +110 |
| 2.50 | 3/2 | +150 |
| 3.00 | 2/1 | +200 |
The format changes nothing in the mathematics of the bet: it is the same price, written three ways.
Implied probability
Any odds can be turned into a probability, called the implied probability:
implied probability = 1 / decimal odds
Odds of 2.10 correspond to an implied probability of 47.62%, and odds of 2.00 to a chance of one in two. The implied probability is a price, not a truth about the event. It combines the organiser's estimate, its margin and, as a rule, adjustments made according to how bettors' stakes are spread.
The average result of a bet, per unit staked, is the real probability of the outcome multiplied by the odds, minus 1. It would be zero if the real probability were equal to the implied one. The margin described below makes the implied probabilities, taken together, larger than the real ones, so the average result of bets is negative for bettors.
The margin built into odds
For an event whose outcomes exclude one another and cover every possibility, the real probabilities add up to exactly 1, that is, certainty. The implied probabilities in an organiser's odds add up to more. The surplus is the margin, also called the "overround".
The example below uses invented odds for a football match with three possible outcomes:
| Outcome | Odds | Implied probability | Without the margin | Fair odds |
|---|---|---|---|---|
| Home win | 2.10 | 47.62% | 45.43% | 2.20 |
| Draw | 3.40 | 29.41% | 28.06% | 3.56 |
| Away win | 3.60 | 27.78% | 26.50% | 3.77 |
| Total | 104.81% | 100.00% |
- The implied probabilities add up to 104.81%. The margin, that is, the surplus over the whole, is 4.81%.
- The "Without the margin" column divides each implied probability by the sum, so that the total becomes exactly the whole. The "Fair odds" column is the inverse of that probability: the odds that would contain no margin.
- If the real probabilities were those in the "Without the margin" column, the organiser would keep on average 4.59% of every stake, whichever of the three outcomes the bettor chose. The calculation is 1 − 1 / sum, and the result is the counterpart of the house edge in casino games.
The proportional split in the table is a simplification. In reality the margin is not necessarily spread evenly across the outcomes, and its size differs from one event to another and from one type of bet to another. The principle stays the same: as long as the implied probabilities add up to more than the whole, the prices contain an edge for the organiser.
Accumulators
An accumulator (a combined bet, a slip with several selections) wins only if every selection is correct. The total odds are the product of the selections' odds, and the probability that all of them are correct, if the events are independent, is the product of their probabilities. The large odds draw attention to the possible amount; the probability falls just as fast as the odds rise.
The margin compounds in the same way. Each selection returns on average only a fraction of the stake, the rest being the organiser's share. On a slip with two selections the fractions multiply; with three selections they multiply three times, and so on. Each selection added thus increases the organiser's expected share of the stake: a slip with many selections carries a far larger total margin than a single bet.
Live betting
Live betting, placed while the event is under way, uses the same formats of odds and the same mathematics. The difference is that the odds are recalculated continuously, according to the score, the time left and other information from the course of the game, and bets may be suspended for a short time after important moments, for example after a goal or when a penalty is awarded. The margin is built into these odds as well.
The pace is different: decisions are made in seconds, and the number of possible bets in a given period grows. Each bet carries its own margin, so the average loss grows with the total amount staked, whatever the pace at which it is staked.
What odds show and what they do not
Odds show the price at which the organiser accepts a bet. They do not show the real probability of an outcome, which is unknown, and they guarantee nothing about the result. Any single bet can end either way, but over a large number of bets the average moves towards what the margin imposes; the mechanism is explained in Variance and the law of large numbers. The link between the margin and the house edge is set out in House edge and RTP.