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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.

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.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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5) For N = 100, what is the smallest k that 3.4 indicates the probability of having k-winners is less than 1? Code or Mathematica to evaluate your answer in 3.4 is fine. 6) For N = 100 and k as in 3.5, argue that there exist possible tournaments with no k-winners. Bonus: Using the Chernoff bound to bound the relevant probabilities, show that for a > 1/2 the probability of there being any aN-winners goes to 0 as N → ∞. Conclude therefore that there exist tournaments without aN-winners, for all sufficiently large N.
Transcribed Image Text:5) For N = 100, what is the smallest k that 3.4 indicates the probability of having k-winners is less than 1? Code or Mathematica to evaluate your answer in 3.4 is fine. 6) For N = 100 and k as in 3.5, argue that there exist possible tournaments with no k-winners. Bonus: Using the Chernoff bound to bound the relevant probabilities, show that for a > 1/2 the probability of there being any aN-winners goes to 0 as N → ∞. Conclude therefore that there exist tournaments without aN-winners, for all sufficiently large N.
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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