Goals in football are few and rare. That fits one of the oldest tools in statistics, the Poissoni distribution, which describes how often a rare event happens in an interval.
The basic model
You give the model an average number of goals for each team in the match, lambda. The chance that a team scores exactly k goals is λ^k × e^(−λ) / k!. With λ = 1.5 the chance of 0 goals is 22.3%, of 1 goal 33.5% and of 2 goals 25.1%.
Do the same for the other team and multiply the two for each scoreline. You get a grid with the probability of every result. Add the cells where the home side scores more, the cells with equal goals, and the cells where the away side scores more, and you have the 1X2. Add by total goals for over/under, and the cells where both teams score for BTTSi.
Where lambda comes from
Lambda comes from one team's attacking strength, the other's defensive strength and home advantage. Strength is estimated from past results and, where available, from expected goals (xGi).
xG adds up the chance of a goal from every shot, based on its location and type. It measures chance quality and not only the score, which makes it steadier than a team's goals over a few matches. A 3-0 with 0.6 xG for is more likely to have been luck than a 1-0 with 2.4 xG for.
What Dixon-Coles adds
In plain Poisson the two teams' goals are independent. Dixon and Coles (1997) showed this fails for low scores: 0-0, 1-0, 0-1 and 1-1 occur at different rates than the independent model says. They added a correction term for those four cells. They also proposed giving older matches less weight so that current form counts for more.
What a model does not do
- It does not see last-minute injuries, manager changes or rotation. That is why we check team news before every tip.
- It knows nothing about motivation, travel or fixture congestion unless you turn them into numbers.
- It carries uncertainty. A 47% is an estimate with an error margin, not an exact value.
The model stays useful as a tool to combine with the market, which is how we use it. See the article on the market and CLV.