Was the cup draw fixed?

In March 2007 three English teams made it through to the quarter-finals of the European Champions League. There was a rumour going around before the draw was made that the draw would be fixed to ensure the English clubs would avoid each other in the draw. Given the very public, and apparently random, nature of the draw this rumour was highly questionable. Nevertheless the English teams were duly drawn apart from each other. Assuming the draw was truly random (and it really is highly unlikely not to have been) what was the probability that this would happen?

The probability is 8/14 - that's better than one in two and so not really much of a coincidence.

Here is why:

In a quarter final there are 8 teams, call them teams A, B, C, D, E, F, G, H of whom 3 are English. We may as well assume teams A, B and C are English.

Now think of the total number of ways (i.e. permuations) in which the draw can be made. There are 8! (8 x 7 x 6...x 2 x 1) because there are 8 ways to draw the first team, then 7 to draw the second etc.

The question then reduces to asking how many of those permutations result in a match between two English teams. To answer this think about the first match drawn (i.e. positions 1 and 2 of the permutation). There are 6 ways the first match can involve two English teams, namely

A, B (meaning A drawn first B drawn second)
B, A
A, C
C, A
B, C
C, B

For EACH of those permutations there are 6! permuations of the resulting 6 positions. So there are 6 x 6! ways that two English teams could be drawn together as the FIRST match.

But, in addition to the first match (i.e. positions 1 and 2) the English teams could also be drawn against each other in 3 other matches, namely matches (3,4) (5,6), and (7,8). So we need to multiply by FOUR, i.e. there are  4 x 6 x 6! ways that two English teams can be drawn together.

Hence the probability of two English teams drawn together is:

(4 x 6 x 6!) / 8!  =  (4 x 6) / (7 x 8) = 6/14

and so the probability of no two English teams being drawn together is one minus that number, which is 8/14.

Norman Fenton

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