Poisson distribution
The Poisson distribution is a probability law that gives the chance of observing 0, 1, 2… rare events when their average is known. In football, it works out the probability of each number of goals for a team from its expected goals (λ), and from there the probability of every exact score.
The formula is P(k) = e^(−λ) × λ^k / k!, where λ is the average number of expected goals and k the number of goals. It suits football well because goals are rare and roughly independent of one another over a match. The statistician Michael Maher applied it to football scores as early as 1982, and it remains the basis of many prediction models.
In practice, you work out each team’s goal distribution, then assume the two are independent: the probability of a score is the product of the two probabilities. Adding up the cells of this grid gives the home win (cells where the home side scores more), the draw (the diagonal), the away win, both teams to score or the total number of goals.
The model has well-known limits. Independence between the two teams is an approximation: in reality, low scores (0-0, 1-1) and draws are slightly more common than a basic Poisson model predicts. In 1997, Mark Dixon and Stuart Coles proposed a correction for these low scores, along with a weighting that gives more importance to recent matches. Above all, everything depends on λ: a Poisson model is only as good as the expected goals it is fed.
Example
With λ = 1.5 expected goals for team A: P(0) ≈ 22.3%, P(1) ≈ 33.5%, P(2) ≈ 25.1%, P(3) ≈ 12.6%. With λ = 1.1 for team B: 33.3%, 36.6%, 20.1% and 7.4%. Probability of 1-1: 0.335 × 0.366 ≈ 12.3%, the most likely score of this match; probability of 0-0: 0.223 × 0.333 ≈ 7.4%.
How Elofoot uses it
It is the engine behind Elofoot's numbers. Expected goals start from each team's actual average xG (failing that, 1.45 at home and 1.15 away), are adjusted for relative strength (league position, form, division strength) and home advantage (× 1.12 / ÷ 1.12), then capped between 0.25 and 4. An independent grid of 0 to 10 goals per team then gives the 1X2, the most likely score, both teams to score and over/under 0.5 to 3.5 goals. During a match, the grid is recalculated over the remaining time from the actual score.
Frequently asked questions
- Why is the Poisson distribution used in football?
- Because goals are rare and roughly independent events, so the number scored in a match fits a Poisson distribution fairly well. You then only need each team’s expected goals to get the probability of every score.
- How do you calculate an exact score probability with Poisson?
- Work out the probability of each team scoring the required number of goals, then multiply the two. With 1.5 and 1.1 expected goals, 1-1 comes out at 33.5% × 36.6% ≈ 12.3%.
- What are the limits of the Poisson model?
- It assumes the two teams’ goals are independent and slightly underestimates draws and low scores, which the Dixon-Coles model (1997) corrects. And it depends entirely on how good the expected-goals estimates are.
See also
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