### 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

Read sections 2 and 3 from chapter 5. Also, complete the questions at the end of each section. Section 2 talks about experiments for which outcomes are equally likely to occur and also discusses the frequency approach to assign probabilities. Section 3 focuses on the concept of events and also touches upon the issue of conditional probability.

Study chapter 3 to learn about basic concepts of probability. Section 1 discusses spaces, events, and their probabilities using many examples. Section 2 elaborates on sets operations, including complements, intersections, and unions using Venn diagrams. Section 3 introduces conditional probability and talks about independent events. Complete the odd-numbered exercises for each section before checking the answers.

### 2.1.2: Counting Rules

Read section 6 from chapter 5. Also, complete the questions at the end of this section. Section 6 introduces formulas for combinations and permutations, which are useful to compute 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

Read sections 1 and 2 from chapter 4. Section 1 defines discrete and continuous random variables. Section 2 introduces the distributions for discrete random variables. This section also talks about the mean and variance calculations. Complete the odd-numbered exercises for each section before checking the answers.

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.

Read sections 8, 10, and 11 from chapter 5. Also, complete the questions at the end of each section. Section 8 talks about binomial probabilities, discusses how to compute their cumulatives, and introduces the mean and standard deviation. Section 10 introduces the Poisson probability formula. Section 11 defines multinomial outcomes and discusses how to compute probabilities by using the multinomial distribution.

Note: For Section 8, the link to the Binomial Calculator part way down the page may not work. If so, you can instead use this Binomial Calculator.

### 2.2.2: Normal Distribution

Read section 2 from chapter 5. This section talks about standard normal curve and how to compute certain areas underneath the curve. This section also contains numerous exercises and examples. Complete the odd-numbered exercises for this section before checking the answers.

Read sections 3, 4, 6, and 7 from chapter 7. Also, complete the questions at the end of each section. Section 3 briefly talks about the history of both the normal distribution and the central limit theorem, and this section also discusses the relation of normal distributions to errors. Section 4 discusses ways of computing the areas under the normal curve. Section 6 discusses the standard normal distribution and the related areas under the standard normal curve. Regarding the calculation of areas, Section 6 also discusses how to translate from non-standard normal to standard normal. Section 7 addresses how to compute (cumulative) binomial probabilities by using normal approximations.

Watch this video on normal distribution. This video introduces 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

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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.

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