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The $2018$ Champions League quarterfinal draw will take place on Friday, March $16^{th}, 2018$ and I wanted to know what is the likelihood that Barcelona will get paired up with Real Madrid? There are $8$ teams left in the pool so a total of $4$ draws will be made.

I think that the number of ways to pairing Barcelona v Real Madrid is $^8C_2$ but I am stuck on how many possible draws there are. Isn't it $\dfrac{^8C_2}{\text{total # of draws}}$?

If someone can walk me through the solution to this problem that would greatly be appreciated. Probability has always given me trouble...

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2 Answers 2

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I think it would be helpful to think as follows: Given that the Real madrid ball has been drawn, what is the probability that the Barcelona ball will be drawn? Of course it is $1/7$.

By way of more explanation: Count the number of possible quarter finals as follows: fix a team, Real, the number of teams this team can play is then 7, so there are 7 possible quarter final matches Real can play. Real Madrid-Barcelona is one of these potential 7.

It may help to try this tactic with a semifinal and convince yourself.

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  • $\begingroup$ But Real Madrid's ball doesn't have to be the first one picked. I feel like there is some conditional probability involved. $\endgroup$
    – user8463863
    Commented Mar 14, 2018 at 22:17
  • $\begingroup$ indeed, we are exploiting some symmetry in the problem. The probability of any team playing another team should be the same, so it suffices to consider Real Madrid being drawn first. $\endgroup$
    – operatorerror
    Commented Mar 14, 2018 at 22:20
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Barcelona is going to play each of the other $7$ teams with equal probability so the answer is $\frac{1}{7}$

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