Poisson Over/Under and Correct Score Calculation

1. Set the two goal rates

Use rates estimated from information available before the match. This example uses home lambda 1.40 and away lambda 0.90 for illustration only.

2. Calculate each goal distribution

Use P(X=k) = e^(-lambda) x lambda^k / k!. OpenStax provides the formula; Microsoft documents a spreadsheet check.

3. Build score cells

Under independence, multiply row and column probabilities. The 1-0 cell is 0.3452 x 0.4066 = 0.1404. The 2-1 cell is 0.2417 x 0.3659 = 0.0884.

4. Sum the market outcome

OpenStax's Poisson definition establishes the distribution rule used in the calculation below.

Under 2.5 goals consists of totals 0, 1 and 2. For independent Poisson variables, total goals are Poisson with lambda 1.40 + 0.90 = 2.30. The under probability is:

P(0) + P(1) + P(2) = 0.1003 + 0.2306 + 0.2652 = 0.5961.

Over 2.5 is 1 - 0.5961 = 0.4039. Independent fair decimal odds are 1 / 0.4039 = 2.476 for Over and 1 / 0.5961 = 1.677 for Under.

5. Verify the grid

• Sum every retained cell and report tail mass outside the grid.
• Confirm home, draw and away sums equal retained mass.
• Compare the totals shortcut with cell summation.
• Test predictions on later matches.

Dixon and Coles show why football score dependence may require adjustment, particularly in low-score cells.

Worked grid logic

For an Over 2.5 calculation, it is usually easier to sum the complementary cells with total goals 0, 1 or 2, then subtract that sum from 1. The included scorelines are 0-0; 1-0 and 0-1; 2-0, 1-1 and 0-2. Every other scoreline in the grid belongs to Over 2.5.

Do not round each cell before summing. Keep full precision through the calculation and round only the displayed result. If a calculator stops at five goals per team, report the omitted tail or raise the ceiling until the retained probability is sufficiently close to one for the intended precision.

Verification checklist

• Home and away rates are non-negative and tied to a stated data cutoff.
• Each one-team distribution sums to the expected retained mass.
• The two-dimensional grid sum matches the product of those retained masses.
• Under and over partitions do not overlap and reconstruct the grid.
• An independent implementation reproduces test fixtures.
• Edge cases cover zero rates, high rates and invalid input.

The output remains conditional on the input rates and Poisson assumptions. Comparing it with odds adds a second evidence layer: timestamp the price, document margin treatment, and do not describe a positive raw difference as realised profit.

Related resources

Review the Poisson model assumptions before using this calculation or implied probability before comparing a market quote.

Continue learning

• Next guide: Pre-Season Football Statistics
• Related guide: Defensive Football Metrics

Assumptions and limitations

The rates and prices are illustrative and exclude margin. Basic independence, constant rates and accurate inputs may fail. Do not treat a calculated fair price as true odds without out-of-sample evaluation and uncertainty analysis.