**Title: Exploring Connected Graphs with Three Nodes****Objective:** Verify the probability that a graph with n-nodes is connected, specifically when n=3, by listing all possibilities of random graphs with three nodes.**Overview:** In this activity, we explore the concept of graph connectivity using a simple case of three nodes. By examining all possible configurations of edges between these nodes, we aim to understand the probability of forming a connected graph.**Method:**1. **List all Possible Graph Configurations:** - Understand that with three nodes, each pair can either be connected by an edge or not. - There are a total of three pairs: (1,2), (2,3), and (1,3).2. **Possible Configurations:** - No edges: The graph is not connected. - One edge: Only one pair is connected; the graph is disconnected. - Two edges: Two pairs are connected; the graph remains disconnected. - Three edges: All pairs are connected; the graph is fully connected.**Analysis:**- **Connectivity Determination:** Out of all possible configurations, only one (all three edges present) forms a connected graph. - **Probability Calculation:** - Determine the number of configurations that result in a connected graph versus all possible configurations. - Calculate probability as the ratio of connected configurations to total configurations.**Conclusion:** By analyzing these configurations, you can verify if the probability aligns with the theoretical formula for the probability of connectivity for a graph with n=3 nodes. This exercise demonstrates fundamental graph theory concepts and their practical probability implications.---**Graph Visualization:**- If graphs are provided, each graph would typically show nodes as circles and edges as lines connecting them.- Diagrams would depict various combinations of edges between the three nodes, aiding in visual comprehension of connected versus disconnected states. This exercise is a foundational step in understanding more complex graph connectivity scenarios.

Answered: Verify through listing all the…

Verify through listing all the possibilities of random graphs with three nodes that the probability that a graph with n-nodes is connected formula is correct when n=3

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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Question

**Title: Exploring Connected Graphs with Three Nodes**

**Objective:**
Verify the probability that a graph with n-nodes is connected, specifically when n=3, by listing all possibilities of random graphs with three nodes.

**Overview:**
In this activity, we explore the concept of graph connectivity using a simple case of three nodes. By examining all possible configurations of edges between these nodes, we aim to understand the probability of forming a connected graph.

**Method:**

1. **List all Possible Graph Configurations:**
- Understand that with three nodes, each pair can either be connected by an edge or not.
- There are a total of three pairs: (1,2), (2,3), and (1,3).

2. **Possible Configurations:**
- No edges: The graph is not connected.
- One edge: Only one pair is connected; the graph is disconnected.
- Two edges: Two pairs are connected; the graph remains disconnected.
- Three edges: All pairs are connected; the graph is fully connected.

**Analysis:**

- **Connectivity Determination:** Out of all possible configurations, only one (all three edges present) forms a connected graph.

- **Probability Calculation:**
- Determine the number of configurations that result in a connected graph versus all possible configurations.
- Calculate probability as the ratio of connected configurations to total configurations.

**Conclusion:**
By analyzing these configurations, you can verify if the probability aligns with the theoretical formula for the probability of connectivity for a graph with n=3 nodes. This exercise demonstrates fundamental graph theory concepts and their practical probability implications.

---

**Graph Visualization:**
- If graphs are provided, each graph would typically show nodes as circles and edges as lines connecting them.
- Diagrams would depict various combinations of edges between the three nodes, aiding in visual comprehension of connected versus disconnected states.

This exercise is a foundational step in understanding more complex graph connectivity scenarios.

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The question ask for "listing all the possibilities of random graphs with three nodes". The attached document lists the first 3 possibilities. Can you assist with additional possibilities?

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