8. (1.4-18) An eight-team single-elimination tournament is set up as follows: Fo

Question

8. (1.4-18) An eight-team single-elimination tournament is set up as follows: For example, Winner eight students (called A - H) set up a tournament among themselves. The top-listed student in each bracket calls heads or tails when his or her opponent flips a coin. If the call is correct, the student moves on to the next bracket (a) How many coin flips are required to determine the tournament winner? (b) What is the probability that you can predict all of the winners? (c) In NCAA Division I basketball, after the "play-in" games, 64 teams participate in ment to determine the national champion. Considering only the remaining 64 teams, how many games are required to determine the a single-elimination tourna national champion? (d) Assume that for any given game, either team has an equal chance of winning (That is probably not true.) On page 43 of the March 22, 1999, issue, Time claimed that the "mathematical odds of predicting all 63 NCAA games correctly is 1 in 75 million." Do you agree with this statement? If not, why not?

Explanation / Answer

a)
total matches to played = n-1 = 8-1 = 7
hence 7 coin flips are required
b)
probability that one can predict all winners = (1/2)^7
= 1/128

c)
number of games =n-1 = 64-1 = 63
d)
probability = (1/2)^63
=1/(9.22*10^18)

this probability is much less than 1 in 75 million

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