1. The Poisson Distribution
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. The Probability Mass Function (PMF) is:
Where λ is the expected number of occurrences, x is the number of occurrences, and e is Euler's number.
2. Modeling Football Goals
Football goals are relatively rare, independent events that occur within a 90-minute timeframe. Empirical evidence shows that goal scoring closely mirrors a Poisson process, making it the industry standard for baseline predictive models.
3. Calculating Expected Goals (λ)
To use Poisson, you need the expected goals (λ) for each team. This is typically derived from:
- Team Attack Strength
- Opponent Defense Strength
- Home Field Advantage
For example, if the home team averages 1.5 goals and the away team defense concedes 10% more than average, λ_home might be 1.65.
4. The 1X2 Probability Model
Assume λ_home = 1.5 and λ_away = 1.2. We calculate the probability of each team scoring 0, 1, 2, 3, etc., goals independently.
For Home scoring 1: P(1; 1.5) = (e^-1.5 * 1.5^1) / 1! = 0.3347
5. Correct Score Matrix
By multiplying the independent probabilities, we form a matrix. For a 1-0 home win:
This 10.08% probability is one cell in a 6x6 matrix covering scores from 0-0 to 5-5.
6. Over/Under Markets
To find the probability of Over 2.5 goals, you sum the probabilities of all cells in the matrix where Home Goals + Away Goals > 2 (e.g., 2-1, 1-2, 3-0, 2-2).
7. Both Teams to Score (BTTS)
Tip: For BTTS, calculate 1 - (Probability Home = 0 + Probability Away = 0 - Probability 0-0).
8. Limitations
Basic Poisson assumes goals are independent. In reality, a team losing 1-0 pushes harder, altering scoring rates. It also ignores late-game dynamics and red cards.
9. Dixon-Coles Correction
Introduced in 1997, the Dixon-Coles model corrects the basic Poisson distribution's underestimation of low-scoring draws (0-0, 1-1) by applying an inflation parameter (rho) to specific scorelines.
10. Calculator Tool
Build your own match models with our Poisson Calculator.