For a Poisson random variable, the mean and variance both equal lambda. The model describes counts in a fixed interval under its stated conditions (OpenStax Poisson distribution).
The Formula
The Poisson probability of exactly k goals is:
P(K = k) = (e^(-lambda) x lambda^k) / k!
Where:
- lambda is the model's expected count for the period
- k is the specific number of goals you want the probability for
- e is Euler's number (approximately 2.718)
- k! is the factorial of k. OpenStax defines the same Poisson parameters and probability mass function (OpenStax Poisson distribution).
In Microsoft Excel, POISSON.DIST(k, lambda, FALSE) returns the individual probability rather than the cumulative probability (Microsoft POISSON.DIST). Record the rates, formula, precision, and maximum score used so the result can be reproduced.
Practical Football Example
Suppose a purely illustrative model assigns:
- Home goal rate: 2.1
- Away goal rate: 0.8
Using the Poisson formula for an illustrative home-team rate of lambda = 2.1:
| Home goals | Probability |
|---|---|
| 0 | 12.2% |
| 1 | 25.7% |
| 2 | 27.0% |
| 3 | 18.9% |
| 4 | 9.9% |
And for the away team (lambda = 0.8):
| Away goals | Probability |
|---|---|
| 0 | 44.9% |
| 1 | 35.9% |
| 2 | 14.4% |
| 3 | 3.8% |
Under an independent home-away goal-count assumption, the probability of a 2-0 home result is:
27.0016% x 44.9329% = 12.133%
This is the model probability generated by the assumed rates. It does not establish that 2.1 and 0.8 are accurate, that the two counts are independent, or that a quoted correct-score price has positive expected value.
Building a Full Scoreline Grid
Calculating combinations from 0-0 through a chosen maximum creates a scoreline grid. A finite grid is not complete unless it retains the probability above that maximum. Either extend the grid until the omitted tail is negligible for the task or include explicit "home above maximum" and "away above maximum" buckets.
Within a complete grid, sum the appropriate cells:
- Match result (1X2): home goals greater than, equal to, or less than away goals.
- Over/Under goals: Sum all cells where the total exceeds 2.5 for over 2.5, and all cells where the total is 2 or fewer for under 2.5.
- BTTS: Sum all cells where both teams score at least one goal.
Estimating Expected Goals
The rates are model estimates, not observed facts about the next match. A reproducible method should specify its training matches, team-strength parameters, home effect, weighting, promoted-team treatment, lineup information, and update timing. Dixon and Coles provide one documented parameterised football model, while time-ordered validation prevents training on future observations (Dixon and Coles, 1997; scikit-learn TimeSeriesSplit).
Dixon and Coles modelled football scores with attack and defence parameters, a home effect, time weighting, and a low-score dependence adjustment. That paper supports one established approach; it does not make every Poisson implementation equivalent (Dixon and Coles, 1997).
Limitations of the Poisson Model
- Fixed pre-match rates: a basic model does not update for an unexpected lineup, dismissal, injury, or score-state tactical change.
- Dependence: multiplying separate team distributions assumes a joint structure that may fit low scorelines poorly. Dixon-Coles is one correction, not a universal cure.
- Parameter error: small rate changes can materially alter exact-score probabilities.
- Distribution fit: real goal data need not have Poisson mean-variance behaviour for every competition and period.
- Data leakage and drift: fitting and evaluating on the same matches exaggerates performance; validation should preserve time order (scikit-learn TimeSeriesSplit).
- Truncation and rounding: dropping tail cells or multiplying rounded percentages makes totals inconsistent.
Validate probability calibration and scoring rules on later matches before using the output for any decision. The model produces conditional estimates, not certainties.
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Related resources
Continue with Expected Goals (xG): What It Means in Betting for the next part of this topic, or return to Betting Glossary: Every Betting Term Explained in Plain English to compare the other guides in this collection.
Continue learning
- Next guide: Regression to the Mean
- Related guide: Scorecast
- Go deeper: Poisson Football Score Model
Assumptions and limitations
The worked goal probabilities assume a Poisson count with one fixed mean and independent scoring processes where scores are combined. Football data can violate constant-rate, independence, and equal mean-variance assumptions. OpenStax supports the distribution formula; Dixon and Coles illustrate a football-specific adjustment.
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