Poisson Distribution & Correct Score Calculator
Model match outcomes from expected goals (xG), then read off the full correct-score matrix — probabilities and fair odds for every scoreline from 0-0 to 5-5.
Score Matrix
| H\A | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 0 | 6.7% | 8.1% | 4.8% | 1.9% | 0.6% | 0.1% |
| 1 | 10.1% | 12.1% | 7.3% | 2.9% | 0.9% | 0.2% |
| 2 | 7.6% | 9.1% | 5.4% | 2.2% | 0.7% | 0.2% |
| 3 | 3.8% | 4.5% | 2.7% | 1.1% | 0.3% | 0.1% |
| 4 | 1.4% | 1.7% | 1.0% | 0.4% | 0.1% | 0.0% |
| 5 | 0.4% | 0.5% | 0.3% | 0.1% | 0.0% | 0.0% |
18+ where legal. Educational calculator only. Bet sizing outputs are not financial advice.
Correct Score Matrix
The same two lambdas above build a full correct-score grid: P(h:a) = P_home(h) × P_away(a) for every combination from 0-0 to 5-5. Because goals cluster at low counts, most of the probability mass sits in the top-left corner of the grid — the five highlighted cells below are the scorelines a bookmaker would price shortest.
| H\A | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 0 | 6.7% 14.88 | 8.1% 12.40 | 4.8% 20.67 | 1.9% 51.67 | 0.6% 172.22 | 0.1% 717.58 |
| 1 | 10.1% 9.92 | 12.1% 8.27 | 7.3% 13.78 | 2.9% 34.44 | 0.9% 114.81 | 0.2% 478.39 |
| 2 | 7.6% 13.23 | 9.1% 11.02 | 5.4% 18.37 | 2.2% 45.93 | 0.7% 153.08 | 0.2% 637.85 |
| 3 | 3.8% 26.45 | 4.5% 22.04 | 2.7% 36.74 | 1.1% 91.85 | 0.3% 306.17 | 0.1% 1275.70 |
| 4 | 1.4% 70.54 | 1.7% 58.78 | 1.0% 97.97 | 0.4% 244.93 | 0.1% 816.45 | 0.0% 3401.86 |
| 5 | 0.4% 235.14 | 0.5% 195.95 | 0.3% 326.58 | 0.1% 816.45 | 0.0% 2721.49 | 0.0% 11339.53 |
Top number = probability of that exact scoreline. Bottom number = fair (no-vig) decimal odds, 1 ÷ probability. Highlighted cells are the five most likely scores for the lambdas entered above.
Most Likely Scorelines
| # | Score | Probability | Fair odds | Result |
|---|---|---|---|---|
| 1 | 1-1 | 12.1% | 8.27 | Draw |
| 2 | 1-0 | 10.1% | 9.92 | Home |
| 3 | 2-1 | 9.1% | 11.02 | Home |
| 4 | 0-1 | 8.1% | 12.40 | Away |
| 5 | 2-0 | 7.6% | 13.23 | Home |
These are fair odds with no bookmaker margin built in — real correct-score markets carry 15-25% overround, so a book quoting 8.50 on a scoreline priced at 8.00 here is close to fair, while 6.50 on the same line is poor value. Compare the fair odds above to the market price before betting a scoreline, and treat the matrix as a starting point, not a certainty — Poisson still assumes home and away goals are independent, which slightly understates low-scoring draws relative to reality.
Poisson Distribution in Soccer Betting
The Poisson distribution is a probability formula that models the number of events occurring in a fixed interval, given a known average rate. In soccer betting, goals scored by each team follow a near-Poisson distribution — making this one of the most mathematically rigorous tools available for pricing match outcomes independently of bookmaker lines.
The formula: P(k goals) = (λᵏ × e⁻λ) / k! where λ (lambda) is the expected number of goals. If a team averages 1.5 goals per game, the probability of scoring exactly 2 goals is (1.5² × e⁻¹·⁵) / 2! = (2.25 × 0.2231) / 2 ≈ 25.1%. Building a full probability matrix for all score combinations (0-0 through 6-6+) lets you calculate win/draw/loss probabilities and price any scoreline bet.
To set your lambda values, use expected goals (xG) data from the current season. xG adjusts for shot quality, giving a more accurate picture of a team's true attacking output than raw goals. A team with 1.8 xG per game but only 1.2 actual goals is likely to regress upward — the Poisson model using xG captures this where raw goal counts don't.
The critical limitation: Poisson assumes statistical independence between home and away goals. In reality, game state affects scoring — a team leading 2-0 often defends, reducing late goals. This correlation means Poisson slightly overestimates draw probabilities. Professional models adjust with a Dixon-Coles correction, but raw Poisson still outperforms most intuitive approaches to soccer probability estimation.
Poisson Formula
P(k goals) = (λᵏ × e⁻λ) / k! λ = expected goals (lambda) k = number of goals (0, 1, 2, 3...) e = Euler's number ≈ 2.71828 Score probability: P(Home=h, Away=a) = P_h(h goals) × P_a(a goals) Home win = Σ P(h,a) where h > a
Poisson Examples
Home λ=1.8, Away λ=1.1. Poisson gives: Home win ≈ 51%, Draw ≈ 25%, Away win ≈ 24%. Most likely score: 1-1 (≈10.9%), then 2-1 (≈10.7%).
Your Poisson model says Away wins 30%. Bookmaker prices Away at 3.50 (IP=28.6%). Your model edge: +1.4%. Bet if you trust your lambda estimates. Combine with CLV tracking to validate model quality over time.
Frequently Asked Questions
What is the Poisson distribution in soccer betting?
The Poisson distribution models the probability of a specific number of goals occurring, given an expected average. By computing Poisson probabilities for each team independently, you can build a full score matrix and derive win/draw/loss probabilities — creating your own market prices.
Where do I get lambda (expected goals) values?
Use Expected Goals (xG) data from FBref, Understat, or WhoScored. xG weights shots by quality and gives a truer picture than raw goals. For each team, calculate average xG per game over the last 10–15 matches, adjusted for home/away split and opponent quality.
How accurate is the Poisson model?
Against the market, Poisson alone is not enough to beat sharp bookmakers — they use the same model. Its value is in finding discrepancies: your lambda estimates vs. the market's implied lambdas. Combined with team news, recent form, and motivation factors, Poisson provides a quantitative baseline.
What are the limitations of Poisson for soccer?
Poisson assumes independent scoring, which breaks down when game state affects play (a team sitting on a lead reduces attacking output). It also doesn't capture match-specific factors like red cards, injuries, or tactical changes. Dixon-Coles correction addresses the low-score correlation issue.
What is a correct score matrix?
A correct score matrix is a grid of every possible scoreline (0-0, 1-0, 0-1, 1-1... up to a chosen ceiling) with the Poisson-modelled probability of each one, calculated as P(home=h) × P(away=a) using the same two expected-goals inputs used for the 1X2 result. It is the same Poisson model as above, just broken out cell-by-cell instead of summed into win/draw/loss.
How do I turn a correct-score probability into fair odds?
Divide 1 by the probability. A scoreline with a 12% Poisson probability has fair decimal odds of 1 / 0.12 ≈ 8.33. Compare that fair number to the market's correct-score price — if the book offers materially higher odds than your fair number, the model sees value, though correct-score markets carry high vig (commonly 15-25%), so require a bigger edge than a 1X2 or over/under bet before staking.
Why do low scores (1-1, 1-0, 2-1) dominate the matrix?
Because most club and international matches average under 3 total goals, the Poisson probability mass concentrates near each team's rounded-down expected goals. With home λ=1.8 and away λ=1.1, for example, 1-1, 2-1 and 1-0 are typically the three most likely scorelines even though the model technically assigns a (small) probability to every score up to double digits.