HMMT 十一月 2011 · 冲刺赛 · 第 20 题

HMMT November 2011 — Guts Round — Problem 20

[ 10 ] The UEFA Champions League playoffs is a 16-team soccer tournament in which Spanish teams always win against non-Spanish teams. In each of 4 rounds, each remaining team is randomly paired against one other team; the winner advances to the next round, and the loser is permanently knocked out 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? 4 3 2

解析

[ 10 ] The UEFA Champions League playoffs is a 16-team soccer tournament in which Spanish teams always win against non-Spanish teams. In each of 4 rounds, each remaining team is randomly paired against one other team; the winner advances to the next round, and the loser is permanently knocked out 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? 4 Answer: We note that the probability there are not two Spanish teams in the final two is the 5 probability that the 3 of them have already competed against each other in previous rounds. Note that the random pairings in each round is equivalent, by the final round, to dividing the 16 into two groups of 8 and taking a winner from each. Now, letting the Spanish teams be A , B , and C , once we fix 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