The Poisson Distribution in Football: A Worked Example
How two expected-goals figures become a full set of scoreline, 1X2, over/under and BTTS probabilities, step by step, with the limits of the method.
By Adam from Poland · · 6 min read
The Poisson distribution is the workhorse of football prediction. Starting from a single number, how many goals a team is expected to score, it tells you how likely 0, 1, 2 or more goals are. Combine two of them and you get every scoreline, and from those every market. This guide works through one match from start to finish.
The idea
Goals are rare events that can happen at almost any moment of a match. When events are rare and roughly independent, the number that occur in a fixed time follows a Poisson distribution. It has a single input: the average, usually written λ (lambda). In football, λ is a team’s expected goals for the match.
The formula is P(k goals) = λk × e−λ ÷ k!, where e ≈ 2.718 and k! is k × (k−1) × … × 1. You never need to do it by hand, but it helps to know there is nothing hidden in it.
Step 1: goals for each team
Suppose a model expects the home side to score 1.6 goals and the away side 1.1. Poisson turns those averages into a full spread:
| Goals | 0 | 1 | 2 | 3 | 4 | 5+ |
|---|---|---|---|---|---|---|
| Home (λ = 1.6) | 20.2% | 32.3% | 25.8% | 13.8% | 5.5% | 2.4% |
| Away (λ = 1.1) | 33.3% | 36.6% | 20.1% | 7.4% | 2.0% | 0.6% |
Notice that even a team expected to score 1.6 goals fails to score about one time in five, and the single most likely number is 1, not 2. Averages hide a lot of spread.
Step 2: every scoreline
If the two teams’ goals are treated as independent, the chance of a scoreline is the product of the two figures. A 1-1 is 32.3% × 36.6% ≈ 11.8%. Doing that for every pair gives the score matrix. The six most likely scores in this match are:
| Score | 1-1 | 1-0 | 2-1 | 2-0 | 0-1 | 0-0 |
|---|---|---|---|---|---|---|
| Probability | 11.8% | 10.8% | 9.5% | 8.6% | 7.4% | 6.7% |
The most likely single score is a draw, even though the home side is the favourite. That is not a contradiction: the home win is spread across many scorelines, the draw across few. Our guide to the correct score matrix explains how to read the full grid.
Step 3: the markets
Adding up the right cells of the matrix gives every market on a match page:
- 1X2: home win 49.0%, draw 24.9%, away win 26.2% (all scores where home is ahead, level or behind).
- Over 2.5 goals: 50.6% (every score with three or more goals in total).
- Both teams to score: 53.2%, which is simply (1 − 20.2%) × (1 − 33.3%).
Every figure comes from the same two numbers, which is why the markets on a page stay consistent with each other. If a model says the home side is a strong favourite, it cannot also say the away side is likely to keep a clean sheet.
Where simple Poisson falls short
- Low-scoring draws. Real matches produce slightly more 0-0 and 1-1 results than independent Poisson predicts, because teams react to the score. The Dixon-Coles adjustment corrects for this.
- The inputs matter most. Poisson is only as good as the λ values fed into it. Most of the work in a real model goes into estimating attack and defence strength, home advantage and form.
- Game state. A red card, an early goal or a team that only needs a draw changes the scoring rate mid-match. A pre-match model can’t see those.
For how the inputs are estimated, see how football prediction models work.
18+. These figures are information, not betting advice. Please read our responsible gambling page.