From goal rates to a score grid
For a Poisson count with rate lambda:
P(X = k) = e^-lambda x lambda^k / k!
OpenStax documents the distribution and assumptions. Suppose illustrative expected goal counts are 1.4 for Home and 1.0 for Away, and initially assume independent scores.
Selected probabilities are:
- P(Home = 0) = e^-1.4 = 0.2466.
- P(Home = 1) = e^-1.4 x 1.4 = 0.3452.
- P(Away = 0) = e^-1.0 = 0.3679.
- P(Away = 1) = e^-1.0 x 1.0 = 0.3679.
Selected score cells
| Score | Joint probability | Fair decimal odds |
|---|---|---|
| 0-0 | 0.2466 x 0.3679 = 9.07% | 1 / 0.0907 = 11.03 |
| 1-0 | 0.3452 x 0.3679 = 12.70% | 1 / 0.1270 = 7.87 |
| 0-1 | 0.2466 x 0.3679 = 9.07% | 11.03 |
| 1-1 | 0.3452 x 0.3679 = 12.70% | 7.87 |
A complete market needs every supported scoreline plus any catch-all selection. Rounding only a few cells does not produce a valid book.
Football-specific adjustment
Dixon and Coles developed a football score model with Poisson regression and adjustments for low-scoring outcomes in a defined historical sample. That paper supports the need to test football-specific dependence; it does not validate the illustrative rates above or disclose a current bookmaker model.
Pricing and settlement
An operator can transform model probabilities, allocate margin and apply liability or trading decisions. The quoted price may therefore differ from simple fair odds. DraftKings' soccer rules are one current example of why regulation time and market settlement must be identified.
Complete and validate the grid
A selected group of low scores is not a full probability distribution. Choose a maximum goal count, calculate every home-away cell up to that limit and measure the omitted tail probability. If the retained cells sum to 98.5%, the remaining 1.5% must be represented by additional cells, an any-other-score outcome or an explicit truncation adjustment. Renormalizing silently would alter every fair price; OpenStax provides the distribution used in this illustrative grid.
The grid should also reproduce its component markets. Summing every cell where Home goals exceed Away goals gives the model home-win probability; diagonal cells give the draw; the remainder gives away win. Summing cells with total goals above two gives the model probability for Over 2.5. These consistency checks catch indexing and truncation errors before prices are compared; Dixon and Coles provide a football-specific score-model reference.
Validation record
- Data cutoff and method used to estimate each goal rate.
- Formula and any low-score or dependence adjustment.
- Maximum score, tail treatment and rounding policy.
- Out-of-sample scoring rule and evaluation period.
- Market settlement period and catch-all selections.
Dixon and Coles provide one football-specific modelling approach, but a current implementation still needs its own input estimation and validation. A transparent grid is valuable because every probability can be traced; transparency does not guarantee predictive accuracy.
Related resources
Use the correct score definition for settlement or Poisson distribution for a deeper formula guide.
Continue learning
- Next guide: Price and Probability Decision Matrix
- Related guide: Steam Moves
Assumptions and limitations
Rates, independence and score cells are illustrative. A useful model must estimate inputs from time-appropriate data, validate the full probability distribution and disclose truncation. No strategy or performance claim is made.
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