Is there a correct score prediction formula?

There is a formula. It is about a hundred years old, it fits football better than it has any right to, and it will not tell you the score. Those three facts are the whole story, so it is worth understanding what it does give you.

The standard approach treats goals as a rate. A team does not score 1.6 goals; it scores at a pace that averages 1.6 goals over ninety minutes, and any particular match lands somewhere around that. The Poisson distribution turns a rate into the probability of each possible count.

The formula itself

For a team expected to score at rate L, the probability of scoring exactly k goals is:

Do that for the home side, do it for the away side, then multiply the two together to get a scoreline. Multiplying assumes the two scores are independent, which is not quite true (a team that goes two down starts chasing), but it is close enough to be useful.

Worked through on one fixture

Say the home team is expected to score 1.6 and the away team 1.1. That is a fairly ordinary fixture with a moderate home favourite. Run the numbers and you get this:

Add the outcome probabilities up and the same model says home win 49 percent, draw 25 percent, away win 26 percent. That part is genuinely informative. The scoreline part is where people get disappointed.

The uncomfortable number

The single most likely score in that fixture has an 11.8 percent chance. The model's best possible answer is wrong seven times out of eight, and there is no fixture anywhere where the top score climbs much past 15 percent. That is not a flaw in the maths. Football is genuinely that noisy.

So a correct score formula does not produce a prediction. It produces a ranked list of about ten plausible results, all of them unlikely, most of them close together. Anyone selling you a formula that names the score is selling you the ranking with the uncertainty removed.

Where the rate comes from

The formula is the easy half. Getting L right is the hard half, and it is where the real work lives:

• Attack and defence strength. How many a team scores relative to the league average, multiplied by how many their opponent concedes relative to the league average.
• Home advantage. Worth roughly a third of a goal, though less than it used to be. The numbers are here.
• Recency. A season-long average is stale by March. Most models weight recent matches more heavily.
• Everything a table cannot see. A suspended centre back, a Thursday night in another country, a team already safe with nothing to play for.

How to actually use it

In a prediction game, the ranked list is a starting point, not an answer. Two habits get more out of it than anything else.

First, treat the top of the list as the field's likely picks. If 1:1 is the model's favourite, it is also the most crowded call in your league, and being right along with twenty other people moves you up nowhere. That is exactly why rarity exists in our scoring: a correct score nobody else called is worth more than the same score everybody called.

Second, look at the gaps. When 1:0 and 1:1 are three points apart in probability, the model is telling you it has no idea and you should pick on your own read. When the top score is way clear of the rest, the model is confident and you need a specific reason to leave it.

The model gives you a distribution. The game asks for one score. Everything interesting about predicting football happens in the gap between those two sentences.

What a formula will never carry

Motivation, weather, a manager on his last week, a derby where the form book gets ignored for ninety minutes. Rate models are averages, and the matches worth thinking hardest about are the ones where the average is least relevant. That is also why the hardest fixtures to call carry the biggest multipliers in our game.