Most football prediction models start from one idea: goals arrive at a roughly steady rate, and the Poisson distribution is the maths for counting events that happen at a steady rate. Feed it a team's expected goals and it gives you the probability of them scoring 0, 1, 2, 3 and so on.
From scoring rates to scorelines
Estimate how many goals the home side is expected to score (from its attack strength and the opponent's defence, plus home advantage) and do the same for the away side. Poisson turns each into a distribution of goals, and multiplying them gives the probability of every scoreline — 1-0, 2-1, 0-0, and the rest.
From scorelines to a 1X2 line
Add up the scorelines where the home team wins, draws, or loses, and you have fair home/draw/away probabilities. The same grid gives over/under and both-teams-to-score for free. That's a complete fair line from two scoring rates.
Where plain Poisson falls short
It assumes the two teams' goals are independent and it slightly misprices low-scoring games (0-0, 1-0, 1-1). The Dixon-Coles model adds a correction for exactly those scores and a time-decay so recent form counts more — which is why Momus uses a Dixon-Coles fit rather than plain Poisson.
See how Momus builds and uses it in how Momus predicts football matches.