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A single elimination tournament with four players is to be held. A total of three games will be played. In Game 1, the players seeded (rated) first and fourth play. In Game 2, the players seeded second and third play. In Game 3, the winners of Games 1 and 2 play, with the winner of Game 3 declared the tournament winner. Suppose that the following probabilities are known:
P(Seed 1 defeats Seed 4) = 0.8
P(Seed 1 defeats Seed 2) = 0.6
P(Seed 1 defeats Seed 3) = 0.7
P(Seed 2 defeats Seed 3) = 0.6
P(Seed 2 defeats Seed 4) = 0.7
P(Seed 3 defeats Seed 4) = 0.6
a. How would you use random digits to simulate Game 1 of this tournament?
b. How would you use random digits to simulate Game 2 of this tournament?
c. How would you use random digits to simulate the third game in the tournament? (This will depend on the outcomes of Games 1 and 2.)
d. Simulate one complete tournament, giving an explanation for each step in the process.
e. Simulate 10 tournaments, and use the resulting information to estimate the probability that the first seed wins the tournament.
f. Ask four classmates for their simulation results. Along with your own results, this should give you information on 50 simulated tournaments. Use this information to estimate the probability that the first seed wins the tournament.
g. Why do the estimated probabilities from Parts (e) and (f) differ? Which do you think is a better estimate of the actual probability? Explain. |
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