Football feels too chaotic to predict. A deflection, a slip or a late penalty can change everything. Yet one simple piece of maths, the Poisson distribution, does a surprisingly good job of describing how many goals a team scores.
This guide explains why it works, walks through a full example, and shows where it starts to break down. You will not need anything beyond a basic calculator.
What is the Poisson distribution?
The Poisson distribution answers one question: if something happens a certain number of times on average, how likely is it to happen exactly 0, 1, 2 or 3 times in a given period?
It suits events that are rare, can happen at any moment, and do not depend much on when the last one happened. People use it for things like calls arriving at a help desk or typos on a printed page.
You only need one number to use it: the average, known as the expected number of events.
Why goals fit the pattern
Goals share the key features of a Poisson event:
- They are rare. A match has many attacks and touches, but only a small number of them end in a goal.
- There are many chances for one to happen. Each attack has a small chance of producing a goal, and a match contains plenty of attacks. Many small chances added together is exactly the situation Poisson describes.
- They are hard to time. A goal can arrive in any minute, and one goal only partly changes the odds of the next.
None of this means football is pure luck. Better teams have a higher expected number of goals. Poisson simply describes the spread of results around that number, because even a strong side will not score the same amount every week.
The formula
The chance of a team scoring exactly k goals, when it is expected to score λ (the Greek letter lambda), is:
P(k) = (λ^k × e^-λ) / k!
Here, e is a constant of roughly 2.718, and k! means k multiplied by every whole number below it, so 3! is 3 × 2 × 1 = 6. By definition, 0! equals 1.
It looks harder than it is, and a spreadsheet will do the work for you. In Excel or Google Sheets, =POISSON.DIST(2, 1.5, FALSE) gives the chance of exactly 2 goals when 1.5 are expected.
A worked example
The numbers here are made up to show the method. They are not from a real match.
Imagine a home team is expected to score 1.5 goals and the away team 1.0. A model would normally get these figures from each side's attack and defence ratings, plus home advantage.
Putting them through the formula gives:
| Goals | Home team (1.5 expected) | Away team (1.0 expected) | |---|---|---| | 0 | 22.3% | 36.8% | | 1 | 33.5% | 36.8% | | 2 | 25.1% | 18.4% | | 3 | 12.6% | 6.1% | | 4 | 4.7% | 1.5% |
From goals to scorelines
If we assume the two teams' goals are independent, we can multiply the two columns to get the chance of any exact score. For example:
- 1-1: 33.5% × 36.8% = about 12.3%
- 1-0: 33.5% × 36.8% = about 12.3%
- 2-1: 25.1% × 36.8% = about 9.2%
- 0-0: 22.3% × 36.8% = about 8.2%
Even the most likely scorelines here happen only about one time in eight. That is a useful lesson in itself: exact scores are very hard to call.
From scorelines to match results
Build a full grid of every scoreline, then add up the squares you care about:
- Home win: all scores where the home team has more goals, about 49%
- Draw: all level scores, about 26%
- Away win: all scores where the away team has more goals, about 25%
You can answer other questions the same way. Adding up every score with three or more total goals gives about 46% in this example. A half goal advantage in expected goals has turned into nearly a two in one edge in win chances over the away side, while still leaving about a one in two chance that the home team does not win.
Where Poisson breaks down
Poisson is a good starting point, not the full picture. Its main weaknesses are:
- Goals are not fully independent. The simple method treats each team's goals as unrelated, but both sides are playing in the same match. Weather, tempo and tactics affect them together.
- Low scoring draws can be mispriced. Plain Poisson tends to get results such as 0-0 and 1-1 slightly wrong. The Dixon Coles model was designed partly to correct this, with an adjustment for low scores.
- Game state matters. A team that goes ahead may sit back, while a team that falls behind may push forward. The expected rate of goals changes during the match, but the basic model holds it fixed.
- Red cards and injuries change things. A sending off can shift the balance sharply, and the model has no way of knowing about it in advance.
- The inputs may be wrong. The output is only as good as the expected goals figure you start with. If that number is poor, the neat looking percentages will be poor as well.
The takeaway
The Poisson distribution works for football because goals are rare, can come at almost any moment, and result from many small chances. Give it a sensible expected goals figure for each team and it turns those two numbers into a full set of scoreline and match result chances.
Treat what comes out as a reasoned estimate, not a forecast of what will happen. Even a likely result often fails to arrive, and that uncertainty is part of why football stays so watchable.