In the UEFA Euro 2004 playoffs draw 10 national football teams were matched in p

Question

In the UEFA Euro 2004 playoffs draw 10 national football teams
were matched in pairs. A lot of people complained that “the draw was not
fair,” because each strong team had been matched with a weak team (this
is commercially the most interesting). It was claimed that such a matching
is extremely unlikely. We will compute the probability of this “dream draw”
in this exercise. In the spirit of the three-envelope example of Section 2.1
we put the names of the 5 strong teams in envelopes labeled 1, 2, 3, 4, and
5 and of the 5 weak teams in envelopes labeled 6, 7, 8, 9, and 10. We shuffle
the 10 envelopes and then match the envelope on top with the next envelope,
the third envelope with the fourth envelope, and so on. One particular way
a “dream draw” occurs is when the five envelopes labeled 1, 2, 3, 4,5 are in
the odd numbered positions (in any order!) and the others are in the even
numbered positions. This way corresponds to the situation where the first
match of each strong team is a home match. Since for each pair there are
two possibilities for the home match, the total number of possibilities for the
“dream draw” is 25 = 32 times as large.
a. An outcome of this experiment is a sequence like 4, 9, 3, 7, 5, 10, 1, 8, 2,6 of
labels of envelopes. What is the probability of an outcome?
b. How many outcomes are there in the event “the five envelopes labeled
1, 2, 3, 4, 5 are in the odd positions—in any order, and the envelopes labeled
6, 7, 8, 9, 10 are in the even positions—in any order”?
c. What is the probability of a “dream draw”?

Explanation / Answer

a. 1/10!, 10 factorial is to total number of possible outcums b. 5!x5! 5! is the total number of arrangements of 12345 and for each one of these there are 5! arrangements of 678910 c. 5!x5! /10! or 1/252

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