Poisson Distribution Calculator — Football Score Predictor

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, then click "Calculate Probabilities." 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 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.

1. Poisson PMF: P(X = k) = (λ^k × 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
2. Scoreline probability: P(home = h, away = a) = P_home(h) × P_away(a)
3. Match outcomes:Sum scoreline probabilities where h > a (home win), h = a (draw), or h < a (away win)
4. Over 2.5:Sum of P(h, a) where h + a > 2.5 (i.e. h + a ≥ 3)
5. 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.

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 ≈ 11.2%
• Aggregating across all scorelines: Home win ≈ 46%, Draw ≈ 27%, Away win ≈ 27%
• Over 2.5 goals ≈ 42%, BTTS ≈ 47%

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 ≈ 66%, Draw ≈ 19%, Away win ≈ 15%
• Over 2.5 goals ≈ 56%, BTTS ≈ 35%

The low away xG suppresses both the BTTS market and away-win probability. Over 2.5 is driven almost entirely by the home team's goal expectation.