1. Introduction

In this tutorial, we’re going to talk about calculating the odds of a team winning a sports match. We present a football game case study for better comprehension of the process.

Sports results prediction is an exciting and challenging problem due to the inherently unpredictable nature of the sport. There’s a seemingly endless number of potential factors that can affect the results of a sports match.

The unpredictable nature is one of the main reasons that people enjoy the sport. Despite the difficulty of predicting the results of sports matches, various stakeholders, including bookmakers, bettors, and fans, invest in the task.

Moreover, with the emergence of online sports betting, the interest in sports match result prediction has increased. Sports experts and sometimes former players often make predictions on upcoming matches, commonly published in the media. The odds of a team winning offer a type of predictor and source of expert advice regarding sports outcomes.

1.1. Odds Usage Frame

Whereas fixed odds reflect the (expert) predictions of bookmakers, the odds in parimutuel betting markets indicate the combined expectations of all punters, which implies an aggregated expert prediction.

Generally, the field of business involves risk-taking, and the well-known risk in business today is that of finance which hinders the financial growth of a business. Bookmakers tend to predict results and outcomes concerning betting in areas like horse riding or football games to make profits and avoid any financial risk. Profits are bound to be made if these predictions are perfectly made, and this helps bookmakers to avoid any form of loss.

Strategies are set up so that the odds favor the bookmakers if a perfect prediction occurs, which helps lessen any risk in the business.

Football is one of the most popular sports on the planet, and many people have followed it very closely. In recent years, new types of data have been collected for many games in various countries, such as play-by-play data, including information on each shot or pass made in a match. We will use football as a case study to illustrate the calculation of odds in sports.

2. The Math Behind the Odds

The calculation behind the odds helps to determine if a bet is worth pursuing.

Odds serve two purposes:

  • They signal the implied probability of the outcome they are attached to
  • They indicate how much money we could win betting on that outcome

There are three main types of displaying odds:

  • Fractional (British) odds are sometimes written as a fraction, such as \frac{6}{1}, or expressed as a ratio, like six-to-one
  • Decimal (European) odds represent the amount that is won for every \$1 that is wagered
  • Money line (American) odds are accompanied by a plus (+) or minus (-) sign, with the plus sign assigned to the lower probability event with the higher payout

The figure below shows the different odds dispositions:

Types of displaying odds

The bookmaker may manipulate the odds to incentivize bets on a certain side, and the sum of the probabilities for a single event will always surpass 100 percent because the bookmaker takes a cut that is baked directly into the odds.

Odds also reveal how much the bookmaker is charging to take our bet. We may hear bettors refer to this amount as “the juice“, “cut“, or “vig(orish).”

2.1. Calculating Winnings with American Odds

To win \$100 on Real Madrid (favorites), we’ll need to wager \$130. If we wagered \$100 on Barcelona, we’d be set to win \$110. We can use the formula below to calculate the potential winnings for any value we wish to wager.
If we bet \$40 on Real Madrid (-130), the equation would look like this: \frac{130}{100} = \frac{40}{x}
Cross-multiply to get: 130*x = 4000, and then solve for “x”:


x = 30.77

Therefore, if we wager \$40 on Real Madrid we’ll win \$30.77, and the payout would be \$70.77 (the original wager plus the winnings).

If we bet \$40 on Barcelona (+110), our equation would look like this: \frac{100}{110} = \frac{40}{x}

Cross-multiply to get: 100*x = 4400, and then solve for “x”:

x = \frac{4400}{100}

x = 44.00

Therefore, if we wager \$40 on Barcelona we’ll win \$44.00, and the total payout would be \$84.00.

Calculating Implied Probability with American Odds

Implied probability is the likelihood of a particular outcome suggested by the odds. Figuring it out involves converting odds into a percentage, which indicates the likelihood that the event will happen vs. the alternative.

Implied probability is useful because if the estimate of the probability of an event occurring is different than a sportsbook?s, we can and should adjust the bet accordingly.

Say we thought one team had a 60% chance of winning, and it was available at 52.4% implied probability. This would be a smart bet. Of course, if we convert the odds available at the sportsbook into a percentage, they’ll include the “juice” or the “vig.” Factoring in the “vig,” the implied probability of all possible outcomes of a game will be above 100%. This is called “Overround”, and it explains why we need to remove the vig from betting lines if we want a more accurate picture of what the oddsmakers expect to happen in the game.

Calculating implied probability can be a tad tricky, but the formula is simple in theoretical terms:

\frac{Risk}{Return} = Implied \quad Probability

Let’s take a look at what this looks like with American odds. We’ll start with positive odds because they are the easiest to work with.

We use the following Formula with Positive Odds:

Implied \quad Probability = \frac{100}{Positive US odds \quad +\quad 100}

We can calculate Barcelona’s (+110) implied probability of winning the game using this same formula. We can’t use this to calculate Real Madrid’s, however, because they have negative odds.

Let?s put Barcelona’s odds into the formula:

Implied \quad Probability = \frac{100}{110 \quad + \quad 100} = 0.476 \quad or \quad 47.6%

Barcelona, therefore, has a 47.6% chance of winning, according to the bookmaker. If one thinks their chance of winning is higher than that, this bet is worth serious consideration.

How Do Negative Odds Differ?

Here is the formula:

Implied \quad Probability = \frac{Negative US odds}{Negative US odds \quad + \quad 100}

When we use actual numbers, it appears far simpler. Using our example from above, we can calculate Real Madrid?s (-130) implied probability of winning the game. Remember, we can’t use this to calculate Barcelona?s because they have positive odds. Then we have:

Implied Probability =\frac{ -\quad - \quad 130}{-\quad -\quad 130 \quad + \quad 100} = 0.565 \quad or \quad 56.5%

Real Madrid has a 56.5% chance of winning the game, according to the bookmaker.

2.2. Calculating Winnings with Decimal Odds

Determining the payout with decimal odds is straightforward: we simply have to multiply the wager by the odds associated with the team we are betting on. To calculate winnings, subtract the original wager from the payout. Let’s take a look:

Winnings = ( Wager * Decimal \quad Odds ) - Wager

Let’s assume Real Madrid?s odds = 2.40 and Barcelona?s odds = 1.61

If we bet $75 on Real Madrid, the winnings are calculated as follows:

Winning =\/ ( \( 75*2.40 ) -75 = \$105

Conversely, if we bet \$75 on Barcelona, the winning would be:

Winning = ( 75*1.61 ) -75 = \$45.75

Calculating Implied Probability with Decimal Odds

To figure out the implied probability, we have to follow this formula:

Implied \quad probability = \frac{100}{Decimal \quad odds}

So for Real Madrid (2.40), the Implied probability or chances of winning = \frac{100}{2.40}=41.7%, and for Barcelona (1.61), the Implied probability or chances of winning = \frac{100}{1.61} = 62.11%

2.3. Fractional Odds in Action

Let’s say Barcelona receives \frac{8}{13} odds in an upcoming game with Real Madrid. These odds suggest that if this game happened 21 times ( 8 + 13 ), Barcelona would lose 8 games and win 13.

To calculate the implied probability of Barcelona winning the game, we need to take the number of times they are expected to win (13) and divide it by the total number of trials (21). This results in a 61.9% chance Barcelona will win the game.

So, the Implied \quad probability = \frac{Denominator}{Denominator \quad + \quad Numerator}

To calculate Real Madrid?s probability of winning the match, we do the same thing. Let’s say, hypothetically, the odds are \frac{11}{8}, meaning if the game happened 19 times ( 11 + 8 ), Real Madrid would lose 11 times and win 8 times.

Therefore, the probability of Real Madrid winning the game is calculated by dividing 8 (the number of times they would win) by 19 (the total number of trials). Real Madrid has a 42.1% chance of winning the game.

3. The Calculation of Odds

Calculating the odds is like calculating the probability of victory of a team winning the game. However, the odds provide a ratio between the required and complimentary events.

We may define the odds as the probability that the required event will occur divided by the probability that the required event will not occur.

We may write the odds as a sentence with two numbers: “required events to complimentary events”, or one ratio. For example, if the required event is winning, the complimentary event is losing, and a team wins 6 times and loses 21 times, then the odds to win are 6 to 21. We may divide the events by the greatest common divisor (GCD) in this case, 3. Hence, the odds to win are 2 to 7, or \frac{2}{7}, \quad or \quad 0.2857.

3.1. How to Calculate the Odds?

Divide the required events by the complimentary events. This procedure is the easiest way to calculate the odds.
Odds to probability formula:
PA = \frac{A}{A \quad + \quad B}
PB = \frac{B}{A \quad + \quad B}
Probabilities to Odds formula:
A = \frac{PA}{PB} = \frac{PA}{1 \quad - \quad PA}
A – event A, for example, Winning.
B – event B, the complimentary event, for example, losing.
PA – the probability of event A.
PB – the probability of event B.

3.2. What Is the Difference Between Odds and Probability?

The odds provide the ratio between the required events and the complimentary events, while the probability provides the ratio between the required events and the total events.


Winning: 3 events. (Required events)
Losing: 9 events. (Complimentary events)
Total: 12 events. (3 + 9).
Odds = \frac{3}{9} = \frac{1}{3} = 0.3333
P = \frac{3}{12}

A different way of calculating the odds is to use the Poisson Distribution.

4. Conclusion

Odds represent the probability of an event happening. Odds enable the stakeholders in the betting business to work out how much money they might win if their bet wins.

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