# MATH 305 - Winona State University ### Number of Credits: 3 S.H.

Introduction to basic ideas and fundamental laws of probability.

Topics include an introduction to combinatorics, probability axioms, conditional probability, discrete random variables, common discrete distributions, expectation, generating functions, limit theorems, and continuous random variables.

Prerequisite: MATH 165 and MATH 210 or permission of instructor.

Offered every semester.

### 2. Course Outline:

I. Combinatorial analysis

A. Rule of sum and rule of product

B. Sampling: sequences, permutations, combinations, multisets

C. Urn models

II. Axioms of probability

A. Sample space and events

B. Axioms of probability

C. Applications of combinatorics

III. Conditional probability

A. Conditional probabilities

B. Bayes’ Theorem

C. Independence

IV. Discrete random variables

A. Discrete random variables and distribution functions

B. Special cases

1. Binomial

2. Negative binomial

3. Hypergeometric

4. Negative hypergeometric

V.

5. Poisson

6. Multivariate versions of the above

C. Expectation of common discrete random variables.

Discrete random vectors

A. Joint distribution function

B. Independent random variables

C. Conditional distributions

### 7.

VI.

D. Distribution of a function of a random variable

Expectation

A. Properties

B. Expectation of a function of a random variable

C. Conditional expectation

1. Computing expectations by conditioning

2. Computing probabilities by conditioning

D. Generating functions

VII. Limit theorems and inequalities

VIII.

Introduction to continuous distributions

A.

Probability density function (p.d.f.)

B.

C.

Expectation and variance

Common continuous distributions

1. Exponential distribution

2. Normal distribution

IX.

Introduction to Stochastic Processes (time permitting)

A.

Discrete-time Markov chains

B.

Poisson process and the exponential distribution

### Course requirements:

Homework assignments, projects, and group activities.

Completion of any exams and quizzes will also be expected. All requirements will be used to assess students’ knowledge of course material and determine their course grade.

### Course materials:

Textbook, supplemental notes, and online resources.

### Assessment of Outcomes

:

Students will be assessed through their performance on course assignments, projects, exams and quizzes.

### References:

Ross, Sheldon,

A First Course in Probability (7 th

Edition)

, Prentice-Hall, 2005.

Ghahramani, Saeed,

Fundamentals of Probability, with Stochastic Processes (3 rd

Edition)

,

Prentice-Hall, 2005.

Durett, Richard,

The Essentials of Probability

, Duxbury Press, 1994.

Kelly, Douglas,

Introduction to Probability

, Macmillan, 1994.

### Rationale

The major focus of the course of the course is to teach students the foundations of probability. In particular students gain an understanding of probability axioms, methods for calculating probabilities using combinatorial methods, conditional probabilities, discrete & continuous random variables, and expectation. Probability is an important component of the Minnesota Standards of Effective Practice for Beginning Teachers standards. These standards can be viewed at: http://www.revisor.leg.state.mn.us/arule/8710/4600.html

.

The current course, MATH 220 – Combinatorics and Graph Theory, is minimally covering the necessary content for meeting those standards in the professional opinion of our mathematics education faculty. Probability is also essential for the study of statistical theory in STAT 450/460 Mathematical Statistics I & II. Reintroducing a course solely in probability will help strengthen those courses and require less remediation of important probabilistic concepts. Furthermore, in order to prepare students interested in pursuing the actuarial science a stand-alone course in probability is essential to their success on Exam P, which is the first Society of Actuaries professional exam. The “P” in Exam P stands for probability.

As a result of adding this course, MATH 220 – Combinatorics and Graph Theory will be dropped. The small portion of this course on graph theory required for Minnesota

Standards of Effect Practice for Beginning Teachers will be covered in other existing

MTED courses.

### Impact of this Course on other Departments, Programs, Majors, or Minors

1. This course will not increase or decrease the total credits required by a major or minor of any other department.

2. No impact, so no forms are necessary.