England V England in the Champions League
7 March, 2008
Alec Ferguson claimed there is a 40% chance of Manchester United meeting an English club in the quarter finals of the Champions League. Is he right?
Let’s assume Liverpool go through. Manchester United will play one of 7 teams. 3 of them are English, and so the chance of Manchester United meeting an English club is 3/7 = 42.9%.
Actually we estimate that Liverpool have a 91% chance of qualifying for the Quarter Finals. So the best guess chance of Manchester United meeting an English Club is 2.95/7 = 42.1%.
OK, so Alec gets his maths GCSE, but would he be able to answer the following:
What is the chance of two all-English Quarter Finals (assuming Liverpool go through)?
Let’s look at one English club. There were 4 English clubs out of 8. There are 3 English clubs out of 7 that can be picked to play against one English club. Then we need the third English club to be picked against an English club. Now there is only 1 English club out of 5 that can be matched against the third club.
So the overall chance of two all-English quarter finals is (3/7) times (1/5) = 3/35 = 8.6%
What is the chance of zero all-English Quarter Finals (assuming Liverpool go through)?
Let’s look at one English club. They must be matched against one of the 4 non-English clubs (chance = 4/7). Now let’s look at the second English club. They also must be be matched against a non-English club. There are three out of five to choose from (chance = 3/5). The third English club must be matched against a non-English club – two out of three of the remaining clubs are non-English (chance = 2/3).
Phew. The good news is because we’ve dealt with three Quarter Finals the last (English V non-English) Quarter Final is fixed. So the chance of zero all-English quarter finals is (4/7) times (3/5) times (2/3) = 8/35 = 22.9%.
What is the chance of exactly one all-English Quarter Final (assuming Liverpool go through)?
Easy: Either there are zero, one or two all-English matches. We already calculated zero and two all-English Quarter Finals. The chance of exactly one all-English quarter final is simple 1 – 3/35 – 8/35 = 24/35 = 68.6%.
What is the chance of at least one all-English Quarter Final (assuming Liverpool go through)?
Again, that’s easy. If we want at least one all-English Quarter Final, that can either be exactly one all-English Quarter Final or exactly two all-English Quarter Finals. So the chance of at least one is 24/35 + 3/35 = 27/35 = 77.1%.
Now, do you think Alec has managed to do all those calculations? He probably didn’t bother because he wouldn’t like to assume Liverpool going through…
Thanks to Nagulan Saravanamuttu, who helped calculate the probabilities for a much easier problem last year when there were three English clubs left in the Champions League.
