Poisson distribution in betting is used to calculate the frequency of any occurrence in a game. In this article, you will learn how to calculate the probability of any score in football, and how to use it to calculate who is likely to win.
What is Poisson Distribution in Betting?
Poisson distribution was developed by 19th century French mathematician Siméon Denis Poisson. It is a probability theory that uses historical sports data to predict the outcome of a sports event. It measures the likelihood of how many times an event will occur during a specific period.
This may seem complicated to someone who has no background in maths, but it is actually a fairly simple method. To put it simply in terms of football betting, Poisson distribution can help you predict how likely each number of goals scored is.
Why is Poisson Distribution Important?
When bookies set their odds, it is important to know how likely any event is, based on past performance. Bookies do not simply come up with odds out of the blue. They use mathematical models. If you want to take a scientific, mathematical approach to betting, you should calculate for yourself how likely you think a specific game event, or set of events will be. That is the first step to finding value. If you have found something that is more likely to happen than what the bookies predict, that is what value is.
Poisson distribution in betting is particularly relevant for games like football, where scoring happens on an incremental scale. It helps you determine the likelihood of each possible score.
Calculating Football Results
The Poisson distribution is commonly used to calculate the likelihood of a specific score in football, as well as a win, lose or draw. You need to first calculate your league’s average goal expectancy, along with the attack strength and defence strength for both sides.
How to calculate goal expectancy
Your team's goal expectancy depends on your team’s attack strength and defence strength, and as well as that of the opposite team.
In our example, we will use the data from the 2018-2019 English Premier League to calculate a hypothetical match between Manchester City and Liverpool. Manchester is the home team, while Liverpool is the away team.
Before calculating these, we need to know:
- The total home goals scored by all EPL teams
- The total away goals scored by all EPL teams
- The average number of home goals and away goals per match for the whole league
We need to calculate Manchester City’s:
- Home goal average
- Average goals allowed per home match
We need to calculate Liverpool’s:
- Away goal average
- Average goals allowed per away match
These stats are easy to find at the Premier League’s official site.
Calculating Attack Strength
With these results, we can easily calculate attack strength for the home and away team. Attack Strength is the team’s average number of goals, divided by the league’s Average number of goals.
Manchester City’s Attack Strength: 3.00 ÷ 1.53 = 1.96
Liverpool’s Attack Strength: 1.78 ÷ 1.147 = 1.55
Calculating Defence Strength
Calculating Defence Strength is just as easy. Simply divide the team’s average number of goals allowed by the league’s average number of goals allowed.
Manchester City’s Defence Strength: 0.63 ÷ 1.147 = 0.55
Liverpool’s Defence Strength: 0.63 ÷ 1.532 = 0.41
Now that we have determined each team’s Attack Strength and Defence Strength, we can calculate each team’s likely score.
Manchester City goal expectancy
To determine how many goals Manchester City will likely score, we need to multiply Manchester City’s Attack Strength by Liverpool’s Defence Strength and the league’s average number of home goals.
That gives us:
1.96 × 0.41 × 1.532 = 1.23
Liverpool goal expectancy
To determine how many goals Liverpool will likely score, we need to multiply Liverpool’s Attack Strength by Manchester City’s Defence Strength and the league’s average number of away goals.
That gives us:
1.55 × 0.55 × 1.147 = 0.997
Average goals scored in the match
Manchester City: 1.23
Using the Poisson Formula to calculate the likelihood of each possible score
Now that we have each team’s home and away defence and attack strengths, we can easily use them with the Poisson formula to calculate the probability of any possible outcome.
The Poisson Formula
The Poisson Formula is:
P (k events in interval) = (λk e –λ) / k!
In this formula:
- P is the probability
- k is the number of occurrences in the interval (number of goals)
- λ is the expected number of goals
- e is Euler's number (e = 2.71828…)
- k! is the factorial of k
Using this formula, you can calculate the probability for any number of goals. However, there are plenty of online calculators which will make the job simpler. To use the calculator, fill in each possible score (limit yourself from 1 to 5) separately in the top in “Event occurrences”, and the expected average goals score per match in the bottom, in “Expected event occurrences”.
That gives us the following probability for Manchester City Goals:
That gives us the following probability for Liverpool City Goals:
Predicting the match outcome based on these probabilities
To get each possible score, simply multiply the probability of each possible score by each team by the probability of each possible score by the other team. This gives you the following distribution:
As you can see, the most likely score is 1 – 1, or 1 – 0 followed by 0 – 0 or 0 – 1. Given the defence averages of both teams, it is easy to see how these would be very likely scores.
How Bookies Convert Estimated Chance Into Betting Odds
Bookies use Poisson distribution to calculate betting odds for outcomes in various markets. You can do the same by converting your calculated probabilities into odds. The calculations are quite simple.
- To calculate the chance of a Manchester City win, we add all the red squares from the table above: that gives us an estimated chance of 0.4142, or 41.42%
- To calculate the chance of a Liverpool win, we add all the green squares from the table above: that gives us an estimated chance of 0.29867, or 29.87%
- To calculate the chance of a draw, we add all the yellow squares from the table above: that gives us an estimated chance of 0.286118, or 28.61%
To convert each of these chances into odds, we use the following formula:
Odds = 1/ (probability)
That gives us the following odds:
- Manchester City win: 1/ (0.4142) = 2.4390
- Liverpool win: 1/ (0.29867) = 3.3333
- Draw: 1/ (0.286118) = 3.4483
You can convert these to American or fractional odds, but decimals are easier to work with. The calculator on our page about implied probability should help you do the maths faster.
Advantages of Poisson Distribution in Betting
Using Poisson distribution in betting has many advantages. First of all, it helps you understand how odds are set in the first place. By adding up the likelihood of various possibilities, bookies are able to set up relatively accurate odds. You can do the same and compare your result to what the bookies are presenting. Betting lines are not only set by using these equations. Popular matches in particular often see the odds offered (betting lines) change, as more money comes in on a particular outcome.
Limitations of Poisson Distribution in Betting
Poisson distribution is a mathematical formula that offers estimated probabilities, not certainties. The more data it has to rely on, the more accurate it can get. On the other hand, no squad is the same for each match of the year.
A player’s injury or absence can make a huge difference in how the entire squad will perform. At the beginning of the season, most teams also have a different line-up than the year before. This makes setting odds using data from a previous season problematic. Still, that does not necessarily put you at a disadvantage, since the bookies also have fewer data to rely on.
As the season goes longer, it becomes easier to predict, since there is more current data available.
How to Create Your Own Poisson Football Spreadsheet
It is not so hard to create your own Poisson distribution calculator with Excel; in fact,you do not need to download one from an external site. This step-by-step guide will show you how to make your own.
1. Calculate your team’s expected goals
First, calculate your team’s expected goals. That is the team's average attack strength × the other team’s defence strength × average goals per match. Below, we calculated Manchester City’s expected goals at 1.23.
2. Create the following table in Excel:
3. Go to the square next to 0, and right click.
4. Click on formulas> Insert Function > Poisson.Dist
5. Fill in:
- X = B5 (or click on the number next to 0)
- Mean = 1.23 (Your team’s expected goals)
- Cumulative = FALSE
6. Move the cursor to the bottom right of C5 and use the plus cursor to drag the formula down.
This gives you the Poisson distribution for 0 to 5 goals of the expected goal average which is 1.23. You can combine the results of your team’s probabilities to get a distribution that looks like this (the same as the above).
TPP Football Stats Centre
Here at ThePuntersPage we have a full range of football statistics that you may also like to check out ranging across all the major countries and leagues:
Poisson Distribution FAQs
Poisson distribution uses probability to determine the odds of any score, based on both team’s past performance and league averages. First, you need to calculate each team’s attack and defence strength and multiply them by the league average. Next, you use the Poisson formula to determine the likelihood of any individual score.
One way to predict football scores is with Poisson distribution. This is a mathematical way to estimate the probability of any score. It is based on both team’s past performance and league averages. Use it to calculate each teams the likelihood of each possible number of goals for a team, and multiply that by the likelihood of each possible number of goals for the other team.
Goal expectancy in football uses the following formula: Attack Strength of the team × Defence Strength of the other team × the league’s Average Number of Goals.
Attack Strength is the team’s average number of goals divided by the league’s Average number of goals for that season.
Using Poisson distribution, the probability of winning a football match is the sum of the probabilities of each individual possible winning score.
To make your own odds, first calculate or estimate the likelihood of an event, then use the following formula: Odds = 1/ (probability). Compare your odds to your bookie's odds to see if they offer any value.
ThePuntersPage Final Say
It can be a bit of work understanding how to calculate odds for various game outcomes. Once you understand Poisson distribution, it becomes much simpler. Luckily, our calculators, as well as the Excel method explained in this article, can help you. Knowing estimated odds and comparing them to the bookies odds is a sure path to finding value in betting.