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Problem 3: Attournamento 0) As a preliminary helper result, show by induction that for events E1, E2,.…..,Em, M P(E, or E2 or ... or EM) < P(Em). (2) m=1 Consider a tournament between N teams, each team playing each of the other teams. 1) How many games will be played in total, in the tournament? We can represent the results of this tournament by a directed graph: node i represents team i, and an edge exists i→ jif team i beat team j. 2) Show (by example) there is a tournament that might occur, where every team is beaten by some team. The above problem suggests that it may be impossible to declare an absolute winner, as everyone may be beaten by somebody. We could relax this slightly in the following way: let's call team i a k-winner if there is a group of k-many teams that were each beaten by team i. Other teams may have beaten team i, but there is at least a group of size k that was roundly beaten by i. 3) If the results of each game are decided by fair coin flip, what is the probability that a given team i is a k-winner? 4) Using result 3.0, bound the probability that there exists a k-winner in a tournament of size N? Write it nicely as you can but don't beat yourself up too much with it.

Problem 3: Attournamento 0) As a preliminary helper result, show by induction that for events E1, E2,.…..,Em, M P(E, or E2 or ... or EM) < P(Em). (2) m=1 Consider a tournament between N teams, each team playing each of the other teams. 1) How many games will be played in total, in the tournament? We can represent the results of this tournament by a directed graph: node i represents team i, and an edge exists i→ jif team i beat team j. 2) Show (by example) there is a tournament that might occur, where every team is beaten by some team. The above problem suggests that it may be impossible to declare an absolute winner, as everyone may be beaten by somebody. We could relax this slightly in the following way: let's call team i a k-winner if there is a group of k-many teams that were each beaten by team i. Other teams may have beaten team i, but there is at least a group of size k that was roundly beaten by i. 3) If the results of each game are decided by fair coin flip, what is the probability that a given team i is a k-winner? 4) Using result 3.0, bound the probability that there exists a k-winner in a tournament of size N? Write it nicely as you can but don't beat yourself up too much with it.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Problem 3: Attournamento 0) As a preliminary helper result, show by induction that for events E1, E2,..., Em, P(E1 or E2 or .. or EM) < P(Em). (2) m=1 Consider a tournament between N teams, each team playing each of the other teams. 1) How many games will be played in total, in the tournament? We can represent the results of this tournament by a directed graph: node i represents team i, and an edge exists i → j if team i beat team j. 2) Show (by example) there is a tournament that might occur, where every team is beaten by some team. The above problem suggests that it may be impossible to declare an absolute winner, as everyone may be beaten by somebody. We could relax this slightly in the following way: let's call team i a k-winner if there is a group of k-many teams that were each beaten by team i. Other teams may have beaten team i, but there is at least a group of size k that was roundly beaten by i. 3) If the results of each game are decided by fair coin flip, what is the probability that a given team i is a k-winner? 4) Using result 3.0, bound the probability that there exists a k-winner in a tournament of size N? Write it nicely as you can but don't beat yourself up too much with it.
Transcribed Image Text:Problem 3: Attournamento 0) As a preliminary helper result, show by induction that for events E1, E2,..., Em, P(E1 or E2 or .. or EM) < P(Em). (2) m=1 Consider a tournament between N teams, each team playing each of the other teams. 1) How many games will be played in total, in the tournament? We can represent the results of this tournament by a directed graph: node i represents team i, and an edge exists i → j if team i beat team j. 2) Show (by example) there is a tournament that might occur, where every team is beaten by some team. The above problem suggests that it may be impossible to declare an absolute winner, as everyone may be beaten by somebody. We could relax this slightly in the following way: let's call team i a k-winner if there is a group of k-many teams that were each beaten by team i. Other teams may have beaten team i, but there is at least a group of size k that was roundly beaten by i. 3) If the results of each game are decided by fair coin flip, what is the probability that a given team i is a k-winner? 4) Using result 3.0, bound the probability that there exists a k-winner in a tournament of size N? Write it nicely as you can but don't beat yourself up too much with it.
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