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)
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.
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