Poisson Distribution Calculator — Football Score Predictor
Calculate football match scoreline probabilities using the Poisson distribution. Enter the expected goals (xG) for each team to instantly see win/draw/win percentages, the five most likely scorelines, over/under 2.5 goals, and both-teams-to-score probabilities.
Odds format
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Average goals expected for home team
Average goals expected for away team
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Most Likely Outcome
Home Win 46.42%
Home Win
46.42%
Draw
25.77%
Away Win
27.81%
| Scoreline | Probability |
|---|---|
| Most likely: 1–1 | 12.26% |
| #2: 1–0 | 11.14% |
| #3: 2–1 | 9.19% |
| #4: 2–0 | 8.36% |
| #5: 0–1 | 8.17% |
Over 2.5 Goals
48.15%
Under 2.5 Goals
51.85%
BTTS (Yes)
51.83%
BTTS (No)
48.17%
| H\A | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 0 | 7.4% | 8.2% | 4.5% | 1.6% | 0.5% | 0.1% |
| 1 | 11.1% | 12.3% | 6.7% | 2.5% | 0.7% | 0.1% |
| 2 | 8.4% | 9.2% | 5.1% | 1.9% | 0.5% | 0.1% |
| 3 | 4.2% | 4.6% | 2.5% | 0.9% | 0.3% | 0.1% |
| 4 | 1.6% | 1.7% | 0.9% | 0.3% | 0.1% | 0.0% |
| 5 | 0.5% | 0.5% | 0.3% | 0.1% | 0.0% | 0.0% |
Rows = Home goals, Columns = Away goals. Highlighted cells are the most likely scores.
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How to Use the Poisson Distribution Calculator
Using this calculator takes two inputs. Enter the home team's expected goals (xG) and the away team's expected goals in the respective fields; the probabilities update as you type. Expected goals represent the average number of goals a team is likely to score based on historical performance, recent form, and head-to-head data.
You can find xG figures from statistics sites such as Understat, FBref, or Opta. A typical Premier League home team averages around 1.5 xG, while an away team averages closer to 1.1. Strong favourites at home may post values above 2.0, while defensive teams may be below 1.0.
The tool outputs three sets of results: the 1X2 match outcome probabilities (home win, draw, away win), the five most likely exact scorelines with individual probabilities, and key betting market probabilities — over/under 2.5 goals and both teams to score (BTTS). All figures are derived mathematically from the Poisson model with no adjustments.
The Formula
The Poisson distribution models the probability of a given number of independent events occurring in a fixed interval. For football, each team's goal tally in a match is treated as an independent Poisson process.
- Poisson PMF: P(X = k) = (λk × e−λ) / k!
- λ = expected goals (the mean of the distribution)
- k = the specific number of goals we're calculating for (0, 1, 2, …)
- k! = factorial of k
- Scoreline probability: P(home = h, away = a) = Phome(h) × Paway(a)
- Match outcomes: Sum scoreline probabilities where h > a (home win), h = a (draw), or h < a (away win)
- Over 2.5: Sum of P(h, a) where h + a > 2.5 (i.e. h + a ≥ 3)
- BTTS: Sum of P(h, a) where h ≥ 1 and a ≥ 1
The model considers all scorelines from 0–0 through 8–8 (an 81-cell matrix). The probability mass beyond 8 goals for either side is negligible for any realistic expected goals value and does not materially affect the results.
Practical Examples
Example 1 — Balanced Match (1.5 xG vs 1.1 xG)
A home team with 1.5 expected goals hosts a side with 1.1 expected goals — a typical mid-table Premier League fixture.
- P(home scores 1) = (1.5¹ × e⁻¹·⁵) / 1! ≈ 0.335
- P(away scores 0) = (1.1⁰ × e⁻¹·¹) / 0! ≈ 0.333
- P(1–0 scoreline) ≈ 0.335 × 0.333 ≈ 11.1%
- Aggregating across all scorelines: Home win ≈ 46%, Draw ≈ 26%, Away win ≈ 28%
- Over 2.5 goals ≈ 48%, BTTS ≈ 52%
The home team is a moderate favourite. The most likely single scoreline is typically 1–0 or 1–1, each around 10–12%.
Example 2 — High-Scoring Favourite (2.5 xG vs 0.8 xG)
A strong home team expected to score 2.5 goals faces a defensive away side likely to manage just 0.8.
- P(home scores 2) = (2.5² × e⁻²·⁵) / 2! ≈ 0.257
- P(away scores 0) = e⁻⁰·⁸ ≈ 0.449
- P(2–0 scoreline) ≈ 0.257 × 0.449 ≈ 11.5% — likely the top result
- Home win ≈ 74%, Draw ≈ 16%, Away win ≈ 10%
- Over 2.5 goals ≈ 64%, BTTS ≈ 50%
The low away xG holds the away-win probability to about 10%; BTTS stays near a coin flip only because the home side is very likely to score. Over 2.5 is driven mostly by the home team's goal expectation.
Frequently Asked Questions
The Poisson distribution models the number of times a random event occurs in a fixed period, given a known average rate. Football goals fit this model well because they are relatively rare, occur independently of each other, and happen at a roughly constant average rate within a match. By treating each team's scoring process as an independent Poisson stream, we can generate a full probability distribution over every possible scoreline.
Expected goals are published by several free and paid data providers. Understat.com provides xG data for the top European leagues going back several seasons. FBref.com (powered by StatsBomb) covers a wide range of leagues. For match previews, sites like SofaScore and WhoScored publish pre-match xG estimates based on recent form and strength of schedule. Many bettors average the season-long xG per game for each team after filtering out extreme outliers.
The Poisson model is a solid baseline and is widely used in quantitative sports betting. Its main limitation is the independence assumption — in reality, a team that falls behind may push forward and concede more goals, creating score correlation. Extensions such as the Dixon-Coles model add a small-score correlation correction. That said, for over/under and 1X2 markets the basic Poisson model performs well enough to identify value relative to bookmaker odds.
BTTS stands for Both Teams To Score. It is a betting market where you win if both the home and away teams score at least one goal each, regardless of the final result. A 1–0 result does not trigger BTTS; a 1–1 or 2–1 does. The Poisson calculator sums the probability of every scoreline where both h ≥ 1 and a ≥ 1 to give the BTTS Yes probability.
For realistic expected goals values (below 4.0), the probability of either team scoring 9 or more goals in a single match is less than 0.01% and has no meaningful impact on the output probabilities. Capping the grid at 8 goals per side captures over 99.99% of the total probability mass for all standard football matches and keeps the calculation fast and precise.
Sources
- Poisson distribution (Wikipedia)
- M. J. Maher, Modelling association football scores (Statistica Neerlandica, 1982)
- M. J. Dixon and S. G. Coles, Modelling Association Football Scores and Inefficiencies in the Football Betting Market (Journal of the Royal Statistical Society, Series C, 1997)
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