Question

A chess tournament has k levels and 2^k players with skills 1 > 2 > ... >2^k. At each level, random pairs are formed and one person from each pair proceeds to next level. When two opponents play, the one with better skills always wins. What is the probability that players 1 and 2 will meet in the final level?

Answer 1

Explanation:

Player one always wins. So he will be in the final. It is obvious that 2n players are separated to two 2n−1 player subgroups. And each group will have one player reaching the final. For player two to reach the final, he must be in a different subgroup from player one, since any of the remaining players in 2,3,...,2n−1 are likely to be one of the (2n−1−1) players in the same subgroup as player one, or one of the 2n−1 players in the subgroup different from player one, the probability that player two is in a different subgroup from player one and that player one and player two will meet in the final is simply 2n−1/(2n−1). I'm wondering if any expert can explain why 2n−1/(2n−1) holds? Really appreciate it!