please provide complete handwritten solution for Q1 Problem 3: Attournamento0) 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=1Consider 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 existsi → 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 bysomebody.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 teamsthat were each beaten by team i. Other teams may have beaten team i, but there is at least a group of size k thatwas 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 nicelyas you can but don't beat yourself up too much with it.

1) Given information: A tournament between teams, each team playing each of the other teams.

Answered: Consider a tournament between N teams,…

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?

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter8: Sequences, Series,and Probability

Section8.6: Counting Principles

Problem 74E: Lottery Powerball is a lottery game that is operated by the Multi-State Lottery Association and is...

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Question

please provide complete handwritten solution for Q1

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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