Video:
Independent events; probability of an intersection  (time: 8:03)

Tree diagrams (time: 6:07)
See the whole entry

## Homework Hints 13-5

Problems 1-6
Use the tree diagram to find each requested probability. See Example 1.

Problem 7-12
Use a tree diagram and Bayes’ theorem. See Example 4.

Problem 13-24
Use the Venn diagram and look for the given set. See Example … See the whole entry

## Homework Hints 13-4

Problem 1
Look in the book to find a statement of this term.

Problems 2-3
Make sure you know these formulas, as well as the conditions for which each applies.

Problem 4
Look in the book to find a statement … See the whole entry

## 13.4 Outline

1. Keno games
2. Independent events
1. independent
2. dependent
3. Probability of an intersection
1. multiplication property of probability
2. birthday problem
4. Probability of a union
2. probability of independent events
5. Drawing with and without replacement
6. Stochastic processes and tree diagrams
See the whole entry

## 12.1 Outline

1. Election problem
1. tree diagram
2. fundamental counting principle
3. arrangement
4. ordered pair/ordered triple
5. define permutation
6. notation for permutations
7. evaluating permutations
2. Factorial
1. definition
2. multiplication property of factorials
3. count-down property
4. permutation formula
3. Distinguishable permutations
1. indistinguishable items
2. distinguishable items
3. formula for distinguishable permutations
See the whole entry

## Individual Research Projects Section 9.3

### Project 9.5

Prepare a classroom demonstration of topology by drawing geometric figures on a piece of rubber inner tube. Demonstrate to the class various ways in which these figures can be distorted.

### Project 9.6

The problem shown in the News … See the whole entry

Videos:
Kruskal’s Algorithm (time: 2:40)

This is a rather foolish introduction to spanning trees  (time: 1:39)
See the whole entry

## Reference Topics 9-2

A historical reference, along with a solution and interesting links for the Konigsberg bridge problem is found at this site:
https://maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem

This site provides a nice discussion of minimum spanning trees.
http://www.people.vcu.edu/~gasmerom/MAT131/mst.html

This is a nice interactive demonstration of Krukal’s … See the whole entry

## Homework Hints 9-2

Problems 1-4
There questions are basically definitions and procedures. Read the book and then paraphrase each in your own words.

Problems 5-12
Review the definition of a tree and see Examples 1 and 2.

Problems 13-20
Review the definition of … See the whole entry

## 9.2 Outline

1. Trees
1. definition
2. spanning tree
2. Minimum spanning trees
1. weight
2. weighted graph
3. definition
4. Kruskal’s algorithm
5. number-of-vertices-and-edges-in-a-tree theorem

## 9.2 Essential Ideas

A tree is a graph which is connected and has no circuits. A tree that is created from another … See the whole entry