As a preliminary helper result, show by induction that for events E1, E2,...,EM, P(E, or E, or... or Em)s ¿ P(EM)
As a preliminary helper result, show by induction that for events E1, E2,...,EM, P(E, or E, or... or Em)s ¿ P(EM)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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P. Answer the following question accordingly. It shouldn’t take that long. Thanks in advance!
![**Title: Understanding Probability in Tournament Graphs**
**Introduction:**
As a preliminary helper result, show by induction that for events \( E_1, E_2, \ldots, E_M \),
\[ P(E_1 \text{ or } E_2 \text{ or } \ldots \text{ or } E_M) \leq \sum_{m=1}^{M} P(E_m) \]
**Tournament Representation:**
Let’s consider a tournament between N teams, with each team playing against every other team.
**Graph Representation:**
- Results of the tournament can be modeled by a directed graph.
- Each node \( i \) represents team \( i \).
- A directed edge \( i \rightarrow j \) exists if team \( i \) beats team \( j \).
**Problem of Winner Declaration:**
- In cases where every team is beaten by some other team, it becomes difficult to declare a winner.
- Solution: Define a "k-winner" as a team \( i \) for which there is a group of \( k \) teams each beaten by team \( i \).
- While other teams might have beaten team \( i \), there exists a subgroup of size \( k \) that was consistently beaten by \( i \).
**Key Question:**
1. If the results of each game are determined by a fair coin flip, what is the probability that a given team \( i \) is a k-winner?
This framework helps in analyzing and predicting potential outcomes in tournaments and determining the likelihood of a team being dominant based on a probabilistic model.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2fbff82e-583d-4003-b329-07736a7bbd92%2F355d7912-b942-441f-8bb4-a70a9be7c106%2Fcb1t4hc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Probability in Tournament Graphs**
**Introduction:**
As a preliminary helper result, show by induction that for events \( E_1, E_2, \ldots, E_M \),
\[ P(E_1 \text{ or } E_2 \text{ or } \ldots \text{ or } E_M) \leq \sum_{m=1}^{M} P(E_m) \]
**Tournament Representation:**
Let’s consider a tournament between N teams, with each team playing against every other team.
**Graph Representation:**
- Results of the tournament can be modeled by a directed graph.
- Each node \( i \) represents team \( i \).
- A directed edge \( i \rightarrow j \) exists if team \( i \) beats team \( j \).
**Problem of Winner Declaration:**
- In cases where every team is beaten by some other team, it becomes difficult to declare a winner.
- Solution: Define a "k-winner" as a team \( i \) for which there is a group of \( k \) teams each beaten by team \( i \).
- While other teams might have beaten team \( i \), there exists a subgroup of size \( k \) that was consistently beaten by \( i \).
**Key Question:**
1. If the results of each game are determined by a fair coin flip, what is the probability that a given team \( i \) is a k-winner?
This framework helps in analyzing and predicting potential outcomes in tournaments and determining the likelihood of a team being dominant based on a probabilistic model.
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