WhamR
Using the Poisson Distribution to Predict Football Scores
A step-by-step worked example of how scoreline probabilities are calculated from expected goals.
Goals in football are rare, roughly independent events — exactly the kind of thing the Poisson distribution was designed to model. It is the backbone of most scoreline models, including the one WhamR uses to rank correct scores.
The formula
If a team is expected to score λ (lambda) goals, the probability it scores exactly k goals is:
P(k) = (λk × e−λ) ÷ k!
Here e is about 2.718 and k! means k × (k−1) × … × 1, with 0! = 1.
A worked example
Suppose a home side is expected to score 1.6 goals and the away side 1.1 goals.
| Goals | Home (λ = 1.6) | Away (λ = 1.1) |
|---|---|---|
| 0 | 20.2% | 33.3% |
| 1 | 32.3% | 36.6% |
| 2 | 25.8% | 20.1% |
| 3 | 13.8% | 7.4% |
From team goals to scorelines
Treating the two teams as independent, the probability of a scoreline is the product of the two columns:
- 1–1: 32.3% × 36.6% = 11.8%
- 1–0: 32.3% × 33.3% = 10.8%
- 2–1: 25.8% × 36.6% = 9.5%
- 2–0: 25.8% × 33.3% = 8.6%
- 0–1: 20.2% × 36.6% = 7.4%
- 0–0: 20.2% × 33.3% = 6.7%
Notice that even the most likely scoreline happens less than 12% of the time. That is why correct-score prices are long and why no model should present a single scoreline as a certainty.
Deriving other markets
The same numbers produce every goals market. The total expected goals is 1.6 + 1.1 = 2.7. The chance of two or fewer goals is about 49.4%, so Over 2.5 is roughly 50.6%. Both teams to score is (1 − 20.2%) × (1 − 33.3%) ≈ 53.2%. One model, many markets — which keeps every pick consistent with every other.
Limitations worth knowing
- Real matches show slightly more draws, especially 0–0 and 1–1, than pure Poisson predicts. Many models apply a small adjustment for this.
- Goals are not fully independent: a red card or an early goal changes how both teams play.
- The output is only as good as the expected goals you feed in.
Key takeaways
- Poisson converts expected goals into probabilities for 0, 1, 2, 3… goals.
- Multiplying the two teams' probabilities gives each scoreline.
- The same table drives totals, BTTS and correct score markets.