### Unit 2: Counting, Probability, and Probability Distributions

How likely is it that a certain event will occur? What are the chances that a given student will receive a grade of 60-69 on their exam? By studying distributions of data, you can determine the probability that a certain event will occur. By looking at the distribution of grades in a class, you can identify the probability that a student will receive between a 60 and 69. Probability is used in business to predicting profits, determine the chances that a business model will affect regulation, and in many other ways.

Before you can focus on probability, you must first learn how to count. What's that? You already know how to count? Maybe – but in this unit, you will learn how to count the different ways that multiple events can occur together. These are called *combinations* and *permutations*, and they are a fundamental concept of probability.

**Completing this unit should take you approximately 12 hours.**

Upon successful completion of this unit, you will be able to:

- identify values of and differentiate between permutations and combinations;
- explain and apply the different methods for determining probability: equally likely outcomes, frequency theory, and subjective theory;
- define and apply the axioms of probability theory;
- apply probability distributions and explain the properties of different distributions;
- solve problems using binomial distribution, and explain when it should be used;
- differentiate between discrete and continuous probability distributions; and
- apply expected value and calculate it for various probability distributions.

- identify values of and differentiate between permutations and combinations;

### 2.1: Counting

Read this introduction to some of the most common formulas and terms in probability.

Watch this lecture, in which Professor Stark works through several examples of counting and probability.

### 2.2: Theories of Probability

Read this section, which explains how to categorize the probability of an event based on what you know about the variables involved.

This section explains some of the most common theories and formulas in probability, and demonstrates set theory by using Venn diagrams.

### 2.3: Set Theory

Watch this video, which demonstrates how to use Venn Diagrams to understand probability.

Watch this video, which discusses Venn diagrams and the addition rule for probability.

Read this section, which covers the symbols used in set theory. For example, the union of two sets A and B is denoted as A∩B. This section also discusses the basic rules of probability and set theory.

Watch these lectures, in which Professor Stark discusses set theory and works through a series of examples.

### 2.4: Probability Fundamentals

Read this section on the probability density function, which is the foundation for how we understand probability.

Watch these lectures, in which Professor Stark works through several examples of how to approach solving problems related to probability.

### 2.5: Probability Distributions and the Binomial Distribution

Watch these videos, which will introduce you to probability distributions and random variables.

Read this chapter, which covers the basic rules of probability and the ways that randomness affects how probabilities are distributed. Make sure you read the introduction and sections 4.1 through 4.4. Attempt the practice problems and homework at the end of each section.

Watch this lecture, in which Professor Stark works through several examples of probability distributions. The second video gives more information on how to find the mean (or expected value) of a discrete random variable.

### Unit 2 Problem Set and Assessment

Choose ten of the problems from this set and solve them. Compare your answers against the given solutions.

Choose ten of the problems from this set and solve them. Compare your answers against the given solutions.

Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment