Guide
Poisson models for football
The Poisson distribution models how often a rare, roughly independent event happens in a fixed interval. Football goals fit the shape well enough that Poisson remains the backbone of most public prediction models.
From expected goals to scorelines
Give each side an expected goals value derived from attack strength, opponent defence and home advantage. The Poisson formula then returns the probability of each exact number of goals, and multiplying the two sides gives every scoreline.
P(k goals) = (λ^k × e^−λ) / k!
Worked example
With λ_home = 1.6 and λ_away = 1.1: P(home scores 1) = 0.323, P(away scores 1) = 0.366, so a 1-1 draw is 0.323 × 0.366 ≈ 11.8%. Summing all scorelines where the home side scores more gives the 1X2 home probability.
Known limitations
Basic Poisson treats the two teams as independent and under-predicts draws and 0-0s. Real matches are correlated: a red card or an early goal changes both sides at once. This is why the models here are blended with an Elo rating and a recent-form component rather than used alone.