HMMT 十一月 2011 · 冲刺赛 · 第 20 题

英文解析

20. [ 10 ] The UEFA Champions League playoffs is a 16-team soccer tournament in which Spanish teamsalways win against non-Spanish teams. In each of 4 rounds, each remaining team is randomly pairedagainst one other team; the winner advances to the next round, and the loser is permanently knockedout of the tournament. If 3 of the 16 teams are Spanish, what is the probability that there are 2
    Spanish teams in the final round?
    Answer: We note that the probability there are not two Spanish teams in the final two is the 4
    probability that the 3 of them have already competed against each other in previous rounds. Note 5
    that the random pairings in each round is equivalent, by the final round, to dividing the 16 into twogroups of 8 and taking a winner from each. Now, letting the Spanish teams be A , B , and C , once wefix the group in which A is contained, the probability that B is contained in this group as well is 7 / 15.
    Likewise, the probability that C will be in the same group as A and B is now 6 / 14. Our answer is thus
    ( ) ( )
    7 6 4
    1 − = .
    15 14 5
    4 3 2