### Unit 2: Elements of Probability and Random Variables

Probabilities affect our everyday lives. In this unit, you will learn about probability and its properties, how probability behaves, and how to calculate and use it. You will study the fundamentals of probability and will work through examples that cover different types of probability questions. These basic probability concepts will provide a foundation for understanding more statistical concepts, for example, interpreting polling results. Though you may have already encountered concepts of probability, after this unit, you will be able to formally and precisely predict the likelihood of an event occurring given certain constraints.

Probability theory is a discipline that was created to deal with chance phenomena. For instance, before getting a surgery, a patient wants to know the chances that the surgery might fail; before taking medication, you want to know the chances that there will be side effects; before leaving your house, you want to know the chance that it will rain today. Probability is a measure of likelihood that takes on values between 0 and 1, inclusive, with 0 representing impossible events and 1 representing certainty. The chances of events occurring fall between these two values.

The skill of calculating probability allows us to make better decisions. Whether you are evaluating how likely it is to get more than 50% of the questions correct on a quiz if you guess randomly; predicting the chance that the next storm will arrive by the end of the week; or exploring the relationship between the number of hours students spend at the gym and their performance on an exam, an understanding of the fundamentals of probability is crucial.

We will also talk about random variables. A random variable describes the outcomes of a random experiment. A statistical distribution describes the numbers of times each possible outcome occurs in a sample. The values of a random variable can vary with each repetition of an experiment. Intuitively, a random variable, summarizing certain chance phenomenon, takes on values with certain probabilities. A random variable can be classified as being either discrete or continuous, depending on the values it assumes. Suppose you count the number of people who go to a coffee shop between 4 p.m. and 5 p.m. and the amount of waiting time that they spend in that hour. In this case, the number of people is an example of a discrete random variable and the amount of waiting time they spend is an example of a continuous random variable.

**Completing this unit should take you approximately 8 hours.**

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

- apply simple principles of probability, and use common terminology of probability;
- calculate conditional probability, and determine whether two events are mutually exclusive and whether two events are independent;
- calculate probabilities using the addition rules and multiplication rules;
- construct and interpret Venn diagrams;
- apply useful counting rules in the context of combinatorial probability;
- identify and use common discrete probability distribution functions;
- calculate and interpret expected values;
- identify the binomial probability distribution, and apply it appropriately;
- identify the Poisson probability distribution, and apply it appropriately;
- identify and use continuous probability density functions; and
- identify the normal probability distribution, and apply it appropriately.

### 2.1: Classical Probability Model

### 2.1.1: Events, Sample Spaces, and Probability

First, we will discuss experiments where outcomes are equally likely to occur and the frequency approach to assigning probabilities. Then, we will focus on the concept of events and touch on the issue of conditional probability.

Read this section about basic concepts of probability, including spaces, and events. This section discusses set operations using Venn diagrams, including complements, intersections, and unions. Finally, it introduces conditional probability and talks about independent events.

### 2.1.2: Counting Rules

This section introduces formulas for combinations and permutations, which are helpful in computing probabilities.

Watch these videos, which introduce Venn diagrams in the context of playing cards and discuss the addition rule for probability.

### 2.2: Random Variables and Distributions

### 2.2.1: Common Discrete Random Variables

This section first defines discrete and continuous random variables. Then, it introduces the distributions for discrete random variables. It also talks about the mean and variance calculations.

Watch these videos on binomial distributions. The first explains how to compute the mean of a binomial distribution. The next two videos introduce binomial probabilities and show how to graph them. The remaining videos elaborate on binomial distribution in the context of basketball examples.

First, we will talk about binomial probabilities, how to compute their cumulatives, and the mean and standard deviation. Then, we will introduce the Poisson probability formula, define multinomial outcomes, and discuss how to compute probabilities by using the multinomial distribution.

### 2.2.2: Normal Distribution

This section talks about the standard normal curve and how to compute certain areas underneath the curve. This section also contains numerous exercises and examples.

First, this section talks about the history of the normal distribution and the central limit theorem and the relation of normal distributions to errors. Then, it discusses how to compute the area under the normal curve. It then moves on to the normal distribution, the area under the standard normal curve, and how to translate from non-standard normal to standard normal. Finally, it addresses how to compute (cumulative) binomial probabilities using normal approximations.

Watch this video on the normal distribution. It introduces the normal distribution and its density curve and explains how to read the areas underneath the normal curve. It also touches on the central limit behavior.

### Unit 2 Assessment

- Receive a grade
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