Question

A single-elimination tournament with four players is to be held. 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 given: \(P(\) seed 1 defeats seed 4\()=.8\) \(P(\) seed 1 defeats seed 2\()=.6\) \(P(\) seed 1 defeats seed 3\()=.7\) \(P(\) seed 2 defeats seed 3\()=.6\) \(P(\) seed 2 defeats seed 4\()=.7\) \(P(\) seed 3 defeats seed 4\()=.6\) a. Describe how you would use a selection of random digits to simulate Game 1 of this tournament. b. Describe how you would use a selection of random digits to simulate Game 2 of this tournament. c. How would you use a selection of random digits to simulate Game 3 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 true probability? Explain.

Short answer

The victory probability of seed 1 in both the 10-tournament and 50-tournament simulations will heavily rely on the simulation experiments.

Step-by-step solution

Game Simulation

For each game, generate a random digit between 0 and 9. If the digit is less than the given probability multiplied by 10, then the higher seed wins. For instance, for Game 1 between seed 1 and seed 4, if the random number is less than 8 (because \(P(\) seed 1 defeats seed 4\()=.8\)), then seed 1 wins.

Tournament Simulation

Utilize the game simulation method for Games 1 and 2. For Game 3, the simulation will be conditional on the winners of the first two games. If seed 1 is among the winners, use the probability that seed 1 will defeat the other winner for simulation. If not, between seed 2 and seed 3, whoever is a winner, use the respective probability for simulation.

10 Tournaments Simulation

Repeat the 'Tournament Simulation' procedure 10 times. Count the number of times the first seed wins. The frequency of seed one's victory divided by 10 gives the estimated probability of victory for seed 1.

50 Tournaments Simulation

Similar to the '10 Tournaments Simulation', conduct it 50 times using data from classmates. Compute the estimated winning probability for seed 1 by dividing the number of seed 1's victories by 50.

Comparing Probabilities

The estimated probabilities from parts '10 Tournaments Simulation' and '50 Tournaments Simulation' may vary because of the variability inherent in random experiments. With an increase in trials (such as from 10 to 50 tournaments), the estimate will be more reliable. Therefore, the estimate from the '50 Tournaments Simulation' should be a better estimate of the true probability.