How to predict a football match with maths: the Poisson model

How many goals will your team score on Saturday? Nobody knows for sure, but statistics can tell you what's most likely. The most used tool for that is a 19th-century formula: the Poisson distribution.

Goals are rare events

A match has ninety minutes and hundreds of plays, but only a few goals. The Poisson distribution describes exactly that: how many times a rare event happens in a fixed interval, when each occurrence is more or less independent of the others. It's used to count calls to a call centre, accidents at a junction or, in our case, goals.

The formula

If a team has an expected average of λ goals (lambda), the probability that it scores exactly k goals is:

P(k) = λ^k · e^−λ / k!

For example, with λ = 1.3, the average goals of a team against an opponent of the same level in GolDraft:

• 0 goals: 27%
• 1 goal: 35%
• 2 goals: 23%
• 3 goals: 10%
• 4 or more: 5%

From team rating to expected goals

The key step is calculating λ for each team. In GolDraft we start from 1.3 expected goals and adjust them by the rating difference between the teams: for every 10 points of difference, the stronger team's expected goals are multiplied by about 1.7 and the weaker team's are divided by the same amount.

An example: an 85-rated team against a 75-rated one. The favourite has about 2.25 expected goals and the opponent about 0.75.

From goals to the result

If we assume each team's goals are independent, the probability of an exact score is the product of the two probabilities. For 85 against 75:

• Probability the favourite scores 1: 24%. Probability the opponent scores 1: 35%. Probability of 1-1: 0.24 × 0.35 ≈ 8%.
• Adding up every score where the favourite scores more, its chance of winning is around 71%, a draw 18% and a defeat 11%.

With two equal teams, on the other hand, each wins 37% of the time and there's a draw 26% of the time. That matches real leagues quite well, where about one in four matches ends level.

The model's limitations

• Goals aren't fully independent. A team that's losing takes more risks, and one that's winning sits back.
• 0-0 and 1-1 draws happen in reality somewhat more often than the simple model predicts. Professional models add corrections for that.
• Ratings aren't everything: injuries, fatigue, home advantage or the weather change a match.

That's why the underdog sometimes wins

Even at 71% in its favour, the favourite fails to win almost three in ten matches. In a four-round cup, that makes reaching the final much harder than it looks: with a 70% chance in each match, winning all four in a row happens only 24% of the time. That's the magic of football, and of the simulator too.