As a preliminary helper result, show by induction that for events El, E2,...,EM, M P(E, or Ex or ... or Em) s 2 P(Em)
As a preliminary helper result, show by induction that for events El, E2,...,EM, M P(E, or Ex or ... or Em) s 2 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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S. Answer the following question accordingly. It shouldn’t take that long. Thanks in advance!
![**Inductive Proof and Tournament Theory**
**Probability of Union of Events**
We begin by establishing a fundamental result via induction. For a series of events \( E_1, E_2, \ldots, E_M \), demonstrate that:
\[
P(E_1 \text{ or } E_2 \text{ or } \ldots \text{ or } E_M) \leq \sum_{m=1}^{M} P(E_m)
\]
This inequality emphasizes that the probability of the union of multiple events is at most the sum of the probabilities of each individual event.
**Tournament Representation and Analysis**
Consider a scenario where \( N \) teams participate in a tournament, with each team competing against every other team.
The outcomes of the tournament can be depicted using a directed graph. In this graph:
- Each node \( i \) symbolizes team \( i \).
- An edge from node \( i \) to node \( j \) (denoted as \( i \rightarrow j \)) exists if team \( i \) beats team \( j \).
**Concept of a k-Winner**
In certain competitions, determining a definitive winner might be challenging, particularly if every team is defeated by at least one other team. To tackle this, we introduce the concept of a "k-winner":
- A team \( i \) is designated as a k-winner if there is a group of \( k \) teams, each of which is defeated by team \( i \).
Despite other teams possibly defeating team \( i \), there exists at least one group of size \( k \) that is consistently defeated by team \( i \).
**Exercise**
Given \( N = 100 \) and the parameter \( k \), argue that there are conceivable tournament arrangements devoid of any k-winners. This exercise reinforces the applicability of directed graphs in understanding tournament dynamics and the concept of k-winners in competitive settings.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2fbff82e-583d-4003-b329-07736a7bbd92%2Fa70e385b-8ff5-4bc9-9390-3151fea42833%2Fnv8n9vd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Inductive Proof and Tournament Theory**
**Probability of Union of Events**
We begin by establishing a fundamental result via induction. For a series of events \( E_1, E_2, \ldots, E_M \), demonstrate that:
\[
P(E_1 \text{ or } E_2 \text{ or } \ldots \text{ or } E_M) \leq \sum_{m=1}^{M} P(E_m)
\]
This inequality emphasizes that the probability of the union of multiple events is at most the sum of the probabilities of each individual event.
**Tournament Representation and Analysis**
Consider a scenario where \( N \) teams participate in a tournament, with each team competing against every other team.
The outcomes of the tournament can be depicted using a directed graph. In this graph:
- Each node \( i \) symbolizes team \( i \).
- An edge from node \( i \) to node \( j \) (denoted as \( i \rightarrow j \)) exists if team \( i \) beats team \( j \).
**Concept of a k-Winner**
In certain competitions, determining a definitive winner might be challenging, particularly if every team is defeated by at least one other team. To tackle this, we introduce the concept of a "k-winner":
- A team \( i \) is designated as a k-winner if there is a group of \( k \) teams, each of which is defeated by team \( i \).
Despite other teams possibly defeating team \( i \), there exists at least one group of size \( k \) that is consistently defeated by team \( i \).
**Exercise**
Given \( N = 100 \) and the parameter \( k \), argue that there are conceivable tournament arrangements devoid of any k-winners. This exercise reinforces the applicability of directed graphs in understanding tournament dynamics and the concept of k-winners in competitive settings.
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