Poisson Distribution explained

The Poisson distribution gives the probability of k events when events arrive independently at a known average rate λ: P(k) = e^−λ·λᵏ/k!. Feed it a team's expected goals and it produces the chance of 0, 1, 2, 3… — and two teams' distributions combine into scorelines, 1X2, totals and BTTS.

It is imperfect in known ways — goals cluster slightly, draws come out a touch under-priced — and every serious football model, including ours, starts from it and corrects from there. Its power is turning one honest input (expected goals) into a full probability surface.