Poisson Calculator: From Expected Goals to Correct Score Odds

The Poisson formula

Poisson distribution football models rest on one formula. If a team is expected to score λ (lambda) goals, the probability that it scores exactly k goals is:

P(k) = λ^k × e^−λ / k!

• λ is the expected goals figure you type in.
• k is the number of goals: 0, 1, 2 and so on.
• e is the constant 2.71828, the base of natural logarithms.
• k! is k factorial: 1 × 2 × … × k, with 0! and 1! both equal to 1.

The probability of a full score is the home team's chance of its goals multiplied by the away team's chance of theirs. A 2–1 is P(home scores 2) × P(away scores 1). The home win is the sum of every score where the home side has more; the draw sums the diagonal, 0–0, 1–1, 2–2 and up. Over 2.5 goals sums every score with three or more goals in total, and both teams to score sums every score where neither side is on nil.

Worked example: 1.6 against 1.1

Take the inputs the Poisson calculator opens with: home expected goals 1.6, away expected goals 1.1. First the zero-goal chances, which need only e^−λ: e^−1.6 is 0.2019, and e^−1.1 is 0.3329.

Each row multiplies the one above by λ and divides by k, which is why the calculator can run all 121 scorelines in an instant.

Now two correct scores. A 1–1 is 0.3230 × 0.3662 = 0.1183, or 11.8%, fair odds 8.45. A 1–0 home win is 0.3230 × 0.3329 = 0.1075, or 10.8%, fair odds 9.30. With these inputs 1–1 is the single likeliest score, even though the home side is the clear favourite overall. That is normal: a favourite wins in many different ways, while a draw at 1–1 is one specific outcome that happens to sit at the peak of both teams' distributions.

Summed across all scores, the same inputs give a home win 49.0% (fair odds 2.04), a draw 24.9% (4.02), an away win 26.2% (3.82), over 2.5 goals 50.6% (1.97) and both teams to score 53.2% (1.88). The correct score predictions in the table run 1–1, 1–0, 2–1, 2–0, 0–1, 0–0, 1–2 and 2–2.

Where expected goals inputs come from

A Poisson calculator is only as good as the two numbers you give it. There are three common starting points, and most serious bettors blend them.

Expected goals (xG). Data providers rate every shot by the chance that a shot from that position and situation becomes a goal, then add the chances up. A team's average xG for and against over recent matches is a steadier guide to its strength than actual goals, which swing with finishing luck and goalkeeping.

Attack and defence ratings. The classic method compares each team with its league average. Attack strength is a team's goals scored per match divided by the league's average goals per match; defence strength is goals conceded per match divided by the league average conceded. A rating of 1.20 means 20% above average, 0.80 means 20% below. The home team's expected goals are then its attack strength × the away team's defence strength × the league's average goals for home teams; the away figure mirrors it with the away average. Home and away averages are kept separate because home sides score more on average in most leagues. You can build the ratings from goals or from xG, and over a whole season or a recent stretch.

Your own adjustments. No rating knows that a striker is out, that a side is rotating for a cup tie or that a fixture is a derby. Adjust the inputs by a tenth or two of a goal for such news, and keep a note of why, so you can check later whether your adjustments helped.

League tables, form and fixtures for the major competitions are on the football betting hub, and the top English division has its own page on Premier League betting.

Comparing fair odds with a sportsbook price

The calculator's odds are fair odds: one divided by the probability, with no margin. A sportsbook's odds carry a margin, so on a well-priced market they will usually be a little shorter than a sound model's fair price. The comparison is useful in two directions.

Spotting value. Say your inputs give the home win 49.0%, fair odds 2.04, and a sportsbook offers 2.20 (an expected price for illustration, not a real quote). The book's price implies 45.5%. By your estimate the bet returns 0.490 × 2.20 = 1.078 per unit staked, an edge of about 7.8%. If the book offers 1.95, the edge is negative and you leave it.

Checking your model. If your fair odds differ from every sportsbook by a wide gap on a big match, the likelier explanation is that your inputs are off, not that the whole market has missed something. Find the difference, usually team news, before betting.

To strip the margin out of a sportsbook's own three-way price and compare like with like, run it through the odds calculator. Sites with the thinnest football margins are compared on the best odds betting sites page.

Known weaknesses of the Poisson model

Independence. The basic model behind any Poisson calculator assumes the home team's goals tell you nothing about the away team's. In real matches they are linked: a goal changes how both sides play.

Draws are underrated. Partly because of independence, the model gives low draws, above all 0–0 and 1–1, slightly less probability than they get in practice. Published refinements, the best known by Dixon and Coles, add a correction for low scores. Without one, treat a draw price close to the fair odds with caution.

Game state. A team leading by two often sits back; a team behind pushes forward and leaves space. Goals do not arrive at a steady rate over 90 minutes, yet the model assumes they do.

Red cards and injuries during the match. A dismissal changes both teams' scoring rates for the rest of the game. A pre-match model cannot see it coming, which matters most for in-play bets placed after the fact.

Thin data. Early in a season, after a transfer window or for a promoted club, ratings rest on few matches or on a different league. The inputs carry more doubt than the neat percentages suggest.

What the calculator does not include

• No bookmaker margin: every price shown is a fair price.
• No team data. You supply both expected goals figures; the tool holds no ratings, xG or fixtures.
• No draw correction or link between the two teams' goals. Each side is modelled on its own.
• No in-play adjustment for the current score, time remaining or red cards.
• No Asian handicap, exact total or half-time markets. The outputs are the three results, over 2.5 goals, both teams to score and the eight likeliest correct scores.
• No stake or returns. For what a bet slip pays, use the bet calculator.

Used with honest inputs, the Poisson calculator is a fast way to turn an opinion about a match into prices you can hold against the market. The model is a starting point; your judgement about the inputs is where the real work is.

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