How long can a run of bad luck last before we start to suspect a football team is actually crap?

I give two simple rules for deciding if a big name team might actually be crap this season. We should conclude that a team is probably crap if they have (1) a run of three losses without a win or (2) establish no clear pattern of winning more than losing after 12 matches.

Anyone who reads newspapers cannot escape the fact that a great deal of what is written revolves around looking at a team’s last few results and then talking about how it reflects the utter brilliance or utter incompetence of the players and/or manager.

Manchesters United and City are apparently incredible this season. Chelsea are apparently lucky in the final minutes. Leicester were apparently a one off. Arsenal are apparently crap. And Liverpool are apparently totally random. That is apparently what we have learnt after two games of competitive football these teams have played this season.

Lets assess these apparent conclusions. For simplicity, lets imagine football teams can either be ‘Brilliant’ or ‘Crap’. These are the two states the media ascribe to teams (although not in exactly those words) so it’s a good starting point. We want to use the logic of probability to know if a certain sequence of results support our team being Crap or Brilliant. Lets denote result sequences in the usual way, e.g. WW for Man U. and LD for Arsenal.

Bayes theorem is a way of using probability theory to work out how a result should effect our beliefs about a team. For this problem we write

The expression P(Crap | W) should be read as follows it is: ‘The probability that my team is Crap given the run of results W”. Similarly, for example, P(W | Brilliant) should be read: ‘‘The probability that my team would have run of results W given it is Brilliant”. Finally, P(Crap) means “before I saw any results what did I think the probability was that my team was Crap”.

All the things on the right-hand side of the equations are things we might be able to give informed guesses about. We want to find out the thing on the left hand side on the basis of the results we seen so far.

I am going to make the following guesstimates:

1, P(W | Brilliant) = 4/5. This is based on the fact that a league winning team wins about 4 out of 5 of the games that end in a result for either team (i.e. ignoring draws).

2, P(W | Crap) = 1/2. Since we are interested in title challengers, winning as many games as you lose is pretty crap. This is what we expect Crap teams to do.

3, P(Crap) = 1-P(Brilliant) = 1/2. This is always the most controversial statement in a Bayes argument. What I am saying here is that there was a 50% chance my team was good and 50% it was bad before the season start.

I encourage you to play about with these numbers and change them a bit, but these are the ones I’ll use in the following example. Lets go!

Are Manchester City and Manchester United Crap?

So given their two wins the Manchester teams have a 72% chance each of being Brilliant.

Are Liverpool Crap?

With one win and one loss are Liverpool looking better or worse than the start of the season?

Despite one of each result, the chance that Liverpool are Crap has increased. But it was 50% to start with in the model, so not something to worry about too much yet.

Are Arsenal Crap?

The draw with Leicester doesn’t count so we get

So it looks like Arsenal are a bit more likely to be Crap than Liverpool.

How long until we can properly call a team crap?

This was my initial question and it depends, of course, on the run of wins and losses. But lets get a few basic rules. Lets imagine that Arsenal lose their next match. Now we have

In this case it is starting to look bad for Wenger. The evidence is mounting that Arsenal are indeed Crap. One more loss after that would make us 94% certain of Crapness.

This gives me rule number 1 for media experts wanting to describe a team as Crap: If a team have three straight losses without a win they are quite likely crap.

How much Klopp craziness should we accept before we start to suspect that Liverpool are Crap? What happens if there is one more win and one more loss?

A similar calculation gives us that the probability a team is crap after WLWLWL is 79%. But it isn’t until after a sequence of 12 WLs, WLWLWLWLWLWL, that the probability rises to 94%.

This gives us rule number two: If a team go 12 matches without winning more often than they lose they are quite likely crap.

It was this rule Liverpool applied last season when they sacked Brendan Rodgers. Boards are often criticised for firing their managers too early in the season, but this argument shows that 12 matches of under-performance or 3 losses in a row is the point to start thinking seriously about problems.