In the draw for the Champion's League knock out phase, out of 16 teams there were 3 Italian teams and 4 English teams. As it turns out, all 3 Italian teams were drawn against English sides. Fabio Capello, the England coach has said that "with all the permutations, that was practically impossible". Is he as good a mathematician as he is a football coach?
Now in a randomised draw, all permutations are equally possible - anyway without knowing much about how the draw took place, that is what I assume. We will calculate how many possible combinations of teams there are. The formula is:
ie the number of ways of choosing 2 from 16, then the number of ways of choosing 2 from 14, etc, then we divide it by 8 factorial because there are 8! different ways of arranging the same match up in order, and we need to remove the duplicate combinations.
With a little bit of algebra manipulation we have
Next we calculate the number of different permutations for there to have the English teams matched with the Italian teams. First, we assign Man U to match 1, Liverpool to match 2, Chelsea to match 3 and Arsenal to match 4. Then we will assign Inter Milan, Juventus and AS Roma to one of these matches.
Inter Milan can choose between 4 matches, Juventus chooses 1 of the other 3, and AS Roma chooses one of the two. Then the last English team chooses 1 of the other 9. So that gives us 4 * 3 * 2 * 9 ways of arranging matches 1 to 4. For matches 5 to 8 we have, according to the calculation of the first part, 8! / (4! 2^4) combinations. So multiplying the two together and cancelling the 4! on the top and bottom, we have:
Then, we take (3) and divide it by (2) and see what we get...
Apparently there is only a 1 in 100 chance, meaning like something like this is a rare occurance, but not like Fabio Capello says, practically impossible.
The conclusion is that Fabio Capello should stick to football coaching and stop pontificating about probability theory.
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