Chapter 8.3, Problem 54E

Elimination Tournaments In an elimination tournament the teams are arranged in opponent pairs for the first round, and the winner of each round goes on to the next round until the champion emerges. The following diagram illustrates a 16-team tournament bracket, in which the 16 participating teams are arranged on the left under Round 1 and the winners of each round are added as the tournament progresses. The top team in each game is considered the “home” team, so the top-to-bottom order matters.

To seed a tournment means to select which teams to play each other in the first round according to their preliminary ranking. For instance, in professional tennis and NCAA basketball the seeding is set up in the following order based on the preliminary rankings: 1 versus 16, 8 versus 9, 5 versus 12, 4 versus 13, 6 versus 11, 3 versus 14, 7 versus 10, and 2 versus 15.³⁵ Exercises 53–56 are based on various types of elimination tournaments.

Someone offers you the following bet: If a randomly chosen seeding of an 8-team tournament results in each team playing a team with adjacent ranking, you win $100; otherwise, you lose $1. What are your expected winnings on this bet? [hinT: See Exercise 38 in Section 8.4.]