## Why Poisson Fits Rare Events
The Poisson distribution models the probability of a given number of events occurring in a fixed interval, when events are rare, independent, and random. Football goals fit these criteria surprisingly well.
## The Poisson Formula
P(X = k) = (e^(−λ) × λ^k) / k!
Where:
- λ = expected number of goals (e.g. 1.5 for a team in a match)
- k = the number of goals you want the probability for
- e ≈ 2.718
**Example:** Team A expected goals = 1.5. What is the probability they score exactly 2?
P(X = 2) = (e^(−1.5) × 1.5²) / 2! = (0.2231 × 2.25) / 2 = 0.2510 = 25.1%
## Building a Full Goals Distribution
For a match where Team A expects 1.3 goals and Team B expects 1.1 goals:
Compute P(A scores 0, 1, 2, 3, 4+) using Poisson with λ = 1.3
Compute P(B scores 0, 1, 2, 3, 4+) using Poisson with λ = 1.1
For every score combination (A=0, B=0), (A=0, B=1), ..., multiply the independent probabilities:
P(A=1, B=0) = P(A scores 1) × P(B scores 0)
Sum all score combinations where A > B for home win probability. Sum all A < B for away win. Sum all A = B for draw.
## The Limitations of Poisson
- Poisson assumes goals are independent — but a goal changes match dynamics (teams often park the bus or push forward), which violates the independence assumption
- Dixon-Coles correction adjusts for the underestimation of 0-0 and 1-0/0-1 scorelines
- For extended coverage, use bivariate Poisson which captures goal correlation between teams
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