Question

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 \(\operatorname{Seed} 2)=0.6\) \(P(\) Seed 1 defeats \(\operatorname{Seed} 3)=0.7\) \(P(\) Seed 2 defeats \(\operatorname{Seed} 3)=0.6\) \(P(\) 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.

Short answer

To simulate the tournament, generate random numbers and use them to decide the winners according to the given probabilities. For multiple simulations, repeat this process and then compute the estimated probabilities. Variables in simulation result in differences between estimated probabilities. Probabilities based on more simulations are usually more accurate.

Step-by-step solution

Simulating Game 1

Create a random number from 0 to 1. If it's less than 0.8 then Seed 1 defeats Seed 4, otherwise Seed 4 wins.

Simulating Game 2

Generate a random number from 0 to 1. Seed 2 wins if the random number is less than 0.6, if not, Seed 3 wins.

Simulating Game 3

The game happens between the winners of Game 1 and Game 2. If Seed 1 won Game 1 and Seed 2 won Game 2, you generate a random number from 0 to 1. If the random number is less than 0.6, Seed 1 wins, otherwise Seed 2 wins. You follow a similar procedure for the other possible matchups.

Simulating the complete tournament

To simulate the tournament, repeat Steps 1, 2 and 3. Record the winner of the tournament.

Simulating multiple tournaments

For simulating multiple tournaments, repeat Step 4 for as many times as required.

Estimating the winning probability

To estimate the winning probability for Seed 1, you calculate the ratio of the number of tournaments won by Seed 1 to the total number of tournaments.

Using classmates' simulations

Combine the tournament results from Step 5 with the results of four classmates and repeat Step 6. This would give the chance of Seed 1 winning based on 50 simulated tournaments.

Comparing probabilities

Explain why the estimated probabilities for Seed 1 winning, from your simulations and the simulations including your classmates' results could be different and state which one you think would be a better estimate of the actual probability. The difference could arise due to the randomness and variability of the simulations. The one based on a larger number of simulations tends to yield a better estimate.