Poisson distribution explained

The Poisson distribution is a mathematical model for rare, random events within a fixed period of time. Goals in football fit it quite well: there are many attacks, but only a few lead to a goal. That is why Poisson has been the basis of many football predictions for decades, including our match previews and the Kicktipp calculator (Kicktipp is a prediction-league platform popular in Germany).

The formula

If λ (lambda) is a team's expected number of goals, the probability of exactly k goals is:

P(k) = λ^k × e^(−λ) / k!

Here e ≈ 2.718 and k! is the factorial (3! = 3 × 2 × 1 = 6). λ can be estimated, for example, from the attacking and defensive strength of both teams, from expected goals or from the odds.

Limits of the model

• Draws: Simple Poisson underestimates draws, especially 0:0 and 1:1. Improved models such as Dixon-Coles correct for this.
• Dependence: Goals are not independent. After falling behind, the match changes.
• Input: The model is only as good as λ. Injured key players, rotation or a change of manager are not automatically included.
• Market: Bookmakers use far more sophisticated models. A pure Poisson model on its own rarely beats the odds in the major leagues.

What it is still useful for

Poisson is simple, transparent and delivers probabilities for all markets in one step: 1X2, over/under, both teams to score and correct scores. It is a very useful tool especially for Kicktipp, where the aim is the most likely score.