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Suppose there is a simple, undirected graph G with 2n vertices and 2n edges, where n ≥ 3. The graph consists of two disjoint cycles with n edges each. For example, if n = 6, the graph would look like this: {fff An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph. An undirected graph that is not connected is called disconnected. The graph defined above and the example show a disconnected graph. (a) A pair of vertices u and v from G is selected uniformly at random from the pairs of distinct vertices with no edge between them. A new graph G' is constructed to be the same as G, except that there is an edge between u and v. What is the probability that G' is connected? (b) k pairs of vertices from G are selected uniformly at random from the pairs of distinct vertices with no edge between them. Repetition is allowed; it is possible, for example, that the same pair appears multiple times in the set of k pairs. A new graph G" is constructed to be the same as G, except that there are k new edges: the edges that correspond to the k selected pairs. What is the probability that G" is disconnected? Hint: For k = 1, the sum of your answers to parts (a) and (b) should equal 1.

Suppose there is a simple, undirected graph G with 2n vertices and 2n edges, where n ≥ 3. The graph consists of two disjoint cycles with n edges each. For example, if n = 6, the graph would look like this: {fff An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph. An undirected graph that is not connected is called disconnected. The graph defined above and the example show a disconnected graph. (a) A pair of vertices u and v from G is selected uniformly at random from the pairs of distinct vertices with no edge between them. A new graph G' is constructed to be the same as G, except that there is an edge between u and v. What is the probability that G' is connected? (b) k pairs of vertices from G are selected uniformly at random from the pairs of distinct vertices with no edge between them. Repetition is allowed; it is possible, for example, that the same pair appears multiple times in the set of k pairs. A new graph G" is constructed to be the same as G, except that there are k new edges: the edges that correspond to the k selected pairs. What is the probability that G" is disconnected? Hint: For k = 1, the sum of your answers to parts (a) and (b) should equal 1.

A First Course in Probability (10th Edition)
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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Suppose there is a simple, undirected graph G with 2n vertices and 2n edges, where n ≥ 3. The graph consists of two disjoint cycles with n edges each. For example, if n = 6, the graph would look like this: An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph. An undirected graph that is not connected is called disconnected. The graph defined above and the example show a disconnected graph. (a) A pair of vertices u and v from G is selected uniformly at random from the pairs of distinct vertices with no edge between them. A new graph G' is constructed to be the same as G, except that there is an edge between u and v. What is the probability that G' is connected? (b) k pairs of vertices from G are selected uniformly at random from the pairs of distinct vertices with no edge between them. Repetition is allowed; it is possible, for example, that the same pair appears multiple times in the set of k pairs. A new graph G" is constructed to be the same as G, except that there are k new edges: the edges that correspond to the k selected pairs. What is the probability that G" is disconnected? Hint: For k= 1, the sum of your answers to parts (a) and (b) should equal 1.
Transcribed Image Text:Suppose there is a simple, undirected graph G with 2n vertices and 2n edges, where n ≥ 3. The graph consists of two disjoint cycles with n edges each. For example, if n = 6, the graph would look like this: An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph. An undirected graph that is not connected is called disconnected. The graph defined above and the example show a disconnected graph. (a) A pair of vertices u and v from G is selected uniformly at random from the pairs of distinct vertices with no edge between them. A new graph G' is constructed to be the same as G, except that there is an edge between u and v. What is the probability that G' is connected? (b) k pairs of vertices from G are selected uniformly at random from the pairs of distinct vertices with no edge between them. Repetition is allowed; it is possible, for example, that the same pair appears multiple times in the set of k pairs. A new graph G" is constructed to be the same as G, except that there are k new edges: the edges that correspond to the k selected pairs. What is the probability that G" is disconnected? Hint: For k= 1, the sum of your answers to parts (a) and (b) should equal 1.
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