Sports betting odds: how does it work ? What calculations ?

The odds of a sporting event is a number which defines both the potential gain to be won in case of an exact pronostic, and the probability of winning it.
This is a fundamental and essential data for the gambler, and is calculated and provided by his bookmaker.
Explanations on this page of the operating principle of odds, and of the different odds scoring systems, leading to a better understanding of this very basis in sports betting.

Calculating Payout

The gain won for an exact pronostic is calculated very simply from the odds which appears as a multiplier coefficient:

Total payout = Stake × Odds

Initial stake is multiply by the odds to obtain the total payout (stake + profit) paied when pronostic is correct. The higher the odds, the greater the gain.

Calculation example

In a sport event with the 3 odds 1x2 (for win/draw/loss):

1.45 / 3.5 / 7

which are therefore the 3 odds in the event of victory, draw and defeat.
For example, you wager 10€ on and

you are betting on victory with odds 1.45. If afterall this happen, you win the total payout 10 × 1.45 = 14.5€.
The profit (without initial stake) will then be of 4.5€.
you are betting on a draw with odds 3.5. In this case you are looking for a total payout of 10 × 3.5 = 35€.
The profit will then be 25€.
you bet on defeat of your team with odds 7. In this last case, you expect to win the total payout of 10 × 7 = 70€.
The profit will in that case be of 60€.

These calculations clearly illustrate that the highest odds allow for a much more attractive gain. Without even knowing the teams playing, we also feel that, according to these odds, victory is much more probable (at least in the eyes, and calculations, of the bookmaker who offers these odds), and the bet on defeat more risky.
The odds is thus also an index of probability, and both are closely and mathematically related.

Odds vs. probabilities in sports betting

The higher the odds, the riskier the bet, i.e. the more improbable the outcome and the lower the chances of winning. In fact, apart from a few details (the bookmaker's margin in particular), the odds are calculated both to reflect the probabilities of events and both to balance the chances of winning and the amount of the winnings. All the mathematical details are described in the article "odds & probabilities" which must absolutely be read and correctly understood because it constitutes the very foundation of how sports betting and odds work.
For now on, here, we simply give a few clues:

Examples of odds vs. probabilities

The well-known and fundamental mathematical relationship is that odds and probability are the mathematical reciprocal of each other. This mathematical relationship is detailed, proved mathematically, explained, ... on this dedicated page.
Just some numerical examples to keep in mind, for now,

odds are 10 for one-in-ten chance, i.e. a probability of 10%
odds are 100 for one-in-hundred chance, i.e. a probability of 1%
odds are 2 for a one-in-two or fifty-fifty chance, i.e. a probability of 50%
odds are 1.5 for 2 in 3 chance, i.e. a probability of 66%
odds are 1.1 for 9 in 10 chance, i.e. a probability of 90%
odds are 1.01 for 99 in 100 chance, i.e. a probability of 99%.
The result here is therefore almost certain, and bookmakers do not, in general, offer the possibility of betting with such odds. In general, the minimum odds offered are 1.1.

Fixed-odds vs. pari-mutuel wagering

The odds for an event are given by the bookmaker who can calculate it in two different manners.

Fixed odds

The qualification "fixed" means that the odds displayed when registering the bet will be the one used to determine the winnings. The odds can still evolve, vary and be recalculated by the bookmaker's traders, but it will remain fixed for the bet made at the time it was made.
Fixed odds can, and generally do, fluctuate over time, depending on:

according to the number of bettors who have placed bets on the event (the bookmaker naturally seeks to balance the bets by playing precisely on the odds in order to ensure, for itself, a gain),
according to the latest sports news (an attacker who has just been injured, etc.),
also according to the odds displayed by competing bookmakers: bookmakers seek to gain in attractiveness, and having slightly higher odds than competitors should attract more bettors

In general, and in a simple commercial view, the bookmaker has an interest in attracting as many bettors as possible, and in hoping that as many as possible lose.

Pari-mutuel wagering

Pari-mutuel wagering works in a very different manner from fixed-odds wagering. This principle is especially applied in horse racing.
You wager on the outcome of an event, knowing only the value at that moment of the odds of the bet.
The final odds, which will be applied for all winners, will ultimately be calculated just before the beginning of the sport event based on the betting achieve by all wagerers of the event. If everyone bets on the victory of the same team for example, the odds will be very low for them.
This principle of pari-mutuel betting is more effective for the bookmaker who cannot lose, since he "simply redistributes", in a certain way, the bettors' stakes (taking, of course, his part). The bookmaker, on the other hand, does not play here on the final odds, and in particular has no interest in showing a more attractive odds to attract bettors, and he also has no interest in a higher number of losers.

Comparing and finding the best odds to win more

Not all bookmakers offer exactly the same odds for the same event. These odds are carefully calculated, notably from mathematical statistics, and the results and estimates of each may differ. Additionally, bookmakers may also be looking for attracting more wagerers by offering higher odds for certain particular events.
A wagerer therefore has every interest in comparing the odds in between bookmakers in order to bet with the highest one: the risk remains strictly the same (it is the same event) but the profit will be greater.
The comparison work can quickly become tedious and boring, and you can either limit yourself to a small number of bookmakers, or use one of the online odds comparators (free or not…).

Main types of betting odds: 3 Different formats

Three alternate ways ca be used for presenting the same odds: decimal (European) odds, fractional (British) odds, and money line (American) odds.
These three main types of betting odds hold no difference in terms of payout, and the chances (of percentage probability) can be easily converted from one to an other format.

European/Decimal Odds

Decimal odds (or "European" odds, also "continental" odds) are, of course, used in Europe, but is also popular in Australia, New Zealand, and Canada. This format is the easier to work with and understand, and also the most adapted numerical format for mathematic calculations. European odds are therefore the format (quasi)exclusively used on this website for all mathematical proofs.
European odds are directly multiply coefficients which in turn makes the total payout calculation the easier way.
When I successfully bet, for example, 20€ according to 1,8 odds, I would receive a total payout of 20 × 1,8 = 36€.
That's total payout, and deducting 20&euros; from this return gives me the 36€ − 20€ = 16€ net profit earned.
British, or fractional, format gives this net profit directly.

British/Fractional Odds

British, or fractional format, odds are used and popular among … british and irish bookmakers.
Odds are then written as a fraction: 2 numbers separated with a slash (or sometimes a hyphen −), for example 5/1, ou 7/2 ou encore 3/10.
With a 5/1 odds, you win 5€ against every 1€ you wager and you get you 1€ back.
Fractionnal odds are also multiply coefficients from which the net profit is easily computed as a multiplication:
For example, when I wager 20€

with 5/1 odds, I would earn 20×5/1 = 100€
with 7/1 odds, I would earn 20×7/2 = 70€
with 3/10 odds, I would earn 20×3/10 = 6€

Conversion to the European decimal format from British fractional format just require adding 1.

American/Money Line Odds

American, or money line, or also "U.S." odds are, of course, used and popular in United States.
Odds for favorites are preceded by a minus − sign and indicate the stake to win 100€ (or $100 ...). On the other hand, odds for the underdogs are preceded by a positive + sign and indicate the amount won for every 100€ staked.
For example, or an other of −200.

american odds of +350: I need to risk 100€ to make a potential win of 350€
american odds of &minus,200: I need here to bet 200€ to win 100€ (net profit)

Actually, odds could always be written preceded by a positive + sign, for example odds of −200 give the same profit as odds of +50.
traditionally, american odds lower than 100 are written negative way.

Odds converter

Fill in a field, odds in any format, and conversions in other formats are calculated automatically.

How does the odds calculator/converter work ?

Simply fill in one of the 4 fields, a decimal (European) odds, fractional (British) odds, or American odds, or even the probability of winning, and all the other fields are calculated and filled automatically.