The 2.5 line: modeling over/under goals with a Poisson distribution (worked example, verified numbers)

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One of the highest-turnover football markets is deceptively simple: will the total goals in a match go over or under 2.5? Where does that line come from, and what is the "fair" probability of each side? Here's the full derivation, with numbers you can reproduce from one formula. A team's goal count in a match can be approximated by a Poisson distribution: P(X = k) = λ^k · e^(−λ) / k! where λ is the team's expected goals (estimated from recent attack strength, opponent defense, home/away).

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