MATH 112 • Final • Discrete Mathematics

Sample Space and Events

Probability

Axioms of Probability

Some Rules

Coin Example

Dice Example

Card Example

Ball Example

Set Example

Birthday Example

Conditioning Events

Total Probability Rule

Example 1

Example 2

Example 3

Example 4

Bayes' Rule

Bayes' Rule Example 1

Bayes' Rule Example 2

Independence

Independence Example 1

Independence Example 2

Random Variables

Probability Mass Function

PMF Example 1

PMF Example 2

Expected Value

Expected Value Example 1

Expected Value Example 2

Variance

Variance Example 1

Variance Example 2

Bernoulli Distribution Part 1

Bernoulli Distribution Part 2

Example 1

Example 2

Binomial Distribution Part 1

Binomial Distribution Part 2

Example 3

Example 4

Geometric Distribution Part 1

Geometric Distribution Part 2

Example 5

Example 6

Negative Binomial Distribution Part 1

Negative Binomial Distribution Part 2

Example 7

Example 8

Discrete Probability 1

Discrete Probability 2 (Spring 2023)

Discrete Probability 3 (Spring 2023)

Discrete Probability 4 (Spring 2023)

Discrete Probability 5 (Fall 2022)

Discrete Probability 6 (Spring 2023)

Conditional Probability and Independence 1

Conditional Probability and Independence 2

Conditional Probability and Independence 3

Conditional Probability and Independence 4

Bayes' Rule 1

Bayes' Rule 2

Bayes' Rule 3

Bayes' Rule 4

Discrete Random Variables 1

Discrete Random Variables 2

Binomial Distribution 1

Binomial Distribution 2

Binomial Distribution 3

Binomial Distribution 4

Negative Binomial Distribution 1

Negative Binomial - Geometric Distribution

Geometric Distribution 1

Geometric Distribution 2

Introduction

Graph Terminology

Handshaking Theorem

Example 1

Special Graphs

Example 2

Bipartite Graphs

Example 3

Example 4

Complete Bipartite Graph

Matching

Subgraph

Example 5

Subgraph Induced

Edge Contraction

Example 6

Complementary Graph

Adjacency Matrices - Undirected Graphs

Adjacency Matrices - Directed Graphs

Example 1

Example 2

Incidence Matrices

Example 3

Isomorphism of Graphs

Example 4

Example 5

Example 6

Example 7

Definition of Paths and Circuits

Connected Graphs

Euler Paths and Circuits

Example 8

Hamilton Paths and Circuits

Example 9

Example 10

Planar Graph

Planar Graph - Euler Formula

Example 11

Corollaries about Planar Graph

Example 12

Introduction

Rooted Tree

Terminology for Rooted Trees

Full m-ary Trees

Some Formulas

Adjacency Matrices

Incidence Matrices

Bipartite Graphs & Euler Circuits (Spring 2023)

Bipartite Graphs, Euler Circuit & Planar Graphs (Fall 2022)

Euler Path, Induced Subgraph & Incidence Matrix (Fall 2022)

Chromatic Number 1

Chromatic Number 2 (Spring 2023)

Isomorphism 1 (Spring 2023)

Isomorphism 2

Isomorphism 3 (Fall 2022)

Isomorphism 4 (Fall 2022)

Isomorphism 5 (Fall 2022)

True/False Problems 1 (Spring 2023)

True/False Problems 2 (Spring 2023)

Induced Subgraphs

Proof for Graphs

Proof for Graphs

Complete Bipartite Graphs

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18