Concept explainers
Elimination Tournaments 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 tournament 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 15.48 Exercises 37–40 are based on various types of elimination tournaments. (Leave each answer as a formula.)
In 2014, after the NCAA basketball 64-team tournament had already been seeded, Quicken loans, backed by investor Warren Buffett, offered a billion dollar prize for picking all the winners.49 An upset occurs when a team beats a higher-ranked team.
a. If you picked the winners completely at random, what is the probability that you would win the prize?
b. If you picked the winners completely at random, what is the probability that your choice would give 15 upsets in the first four rounds? [HinT: See Exercise 67 in Section 7.4.]
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Finite Mathematics
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