Over/Under 2.5 Goals: Calculate the Probability with the Poisson Distribution
How likely are three or more goals? With the Poisson distribution and an expected goal total you can calculate the fair odds yourself.
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Over/under 2.5 goals is one of the most popular football markets. The question is simple: will three or more goals be scored? A classic statistical model gives a good estimate.
The idea. Goals are rare, largely independent events. The Poisson distribution suits such counts. You need only one number: the expected total goals in the match, for example from expected goals (xG) or season averages.
The formula. P(k goals) = e^(-λ) × λ^k / k!, where λ is the expected goal total. For under 2.5, add the probabilities of 0, 1 and 2 goals.
Worked example with λ = 2.7. P(0) = 6.7 %. P(1) = 18.1 %. P(2) = 24.5 %. Under 2.5 is 49.4 %, over 2.5 is 50.6 %. Fair odds for over 2.5 are 1 / 0.506 ≈ 1.98.
Compare with the price. If a bookmaker offers 1.85 on over 2.5, the implied probability is 54 %. According to the model the bet is too expensive. At 2.10 it would be slightly positive.
The limits. Poisson slightly underestimates draws, ignores game-state effects and is only as good as your λ. Red cards, weather or rotation are not captured automatically. Bookmakers use far more refined models.
Bottom line. Poisson is a good tool for judging odds, not a money machine. If you cannot estimate λ better than the market, the formula mostly reveals one thing: the bookmaker's margin.