## Unit 4 Activities

##### Time Advisory

Completing this unit should take approximately 11 hours.- Subunit 4.1: 1 hour
- Subunit 4.2: 1 hour and 30 minutes
- Subunit 4.3: 2 hours and 30 minutes
- Subunit 4.4: 2 hours and 10 minutes
- Subunit 4.5: 3 hours and 50 minutes

##### Learning Outcomes

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

- Identify discrete and continuous random variables.
- Construct the probability distribution of a discrete random variable.
- Use the probability distribution of a discrete random variable to calculate probabilities of individual or multiple outcomes.
- Calculate the mean, standard deviation, and variance of a discrete random variable.
- Calculate the mean and standard deviation for sums and differences of independent random variables.
- Use the binomial distribution to solve statistical problems.
- Use discrete distributions to make real-world decisions.

##### Standards Addressed (Common Core and Advanced Placement):

- CCSS.Math.Content.HSS-IC.A.2
- CCSS.Math.Content.HSS-MD.A.1
- CCSS.Math.Content.HSS-MD.A.2
- CCSS.Math.Content.HSS-MD.A.3
- CCSS.Math.Content.HSS-MD.A.4
- CCSS.ELA-Literacy.RST.11-12.2
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10
- AP III.A.4
- AP III.A.5
- AP III.A.6
- AP III.B.1
- AP III.B.2

**4.1 Random Variables**

*In Unit 1 and Unit 2, you learned how to organize raw data into charts, graphs, and other displays. In Unit 3 you learned the basic ideas of probability. Unit 4, "Random Variables,” combines these ideas of data and probability to create a probability distribution to describe a set of outcomes and their respective probabilities. There is a bit of theory in this unit, but it is not overwhelming, and it is presented in a practical context.*

*This subunit introduces new vocabulary: the difference between a discrete random variable and a continuous random variable. We will concentrate on discrete random variables for much of Unit 4, and then we will study the most famous continuous random variable, the bell curve, in Unit 5.*

**Explanation**: Saylor Academy's Flexbook: Jill Schmidlkofer's *Advanced Probability and Statistics*: "Section 4.1: Two Types of Random Variables”; partially adapted from David Lane's *Online Statistics Education: A Multimedia Course of Study* and CK-12: *Advanced Probability and Statistics* (PDF)

Instructions: Read Section 4.1. This short lesson introduces you to new vocabulary, which you should note: random variable, discrete random variable, and continuous random variable.

Reading this section and taking notes should take approximately 20 minutes.

*Standards Addressed (Common Core and AP):*

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is adapted from works attributed to David Lane and CK-12, which can be found here and here.

**Web Media:** Khan Academy's "Random Variables” (YouTube)

Instructions: Watch the video. This is a basic introduction to the concept of a random variable

Watching this lecture should take approximately 5 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.1
- AP III.A.4
- AP III.A.5

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is attributed to Khan Academy.

**Web Media: **Khan Academy's "Discrete and Continuous Random Variables” (YouTube)

Instructions: Watch the video, which gives you a more in-depth understanding of a random variable. You will learn the difference between a discrete random variable and a continuous random variable.

Watching this lecture should take approximately 15 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.1
- AP III.A.4
- AP III.A.5

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is attributed to Khan Academy.

**Checkpoint:** Saylor Academy's Flexbook: Jill Schmidlkofer's *Advanced Probability and Statistics*: "Section 4.1: Two Types of Random Variables”; partially adapted from David Lane's *Online Statistics Education: A Multimedia Course of Study* and CK-12: *Advanced Probability and Statistics* (PDF)

Instructions: Work on the review questions at the end of Section 4.1. You will use the definitions learned earlier to answer the questions. A short answer key is provided at the end of the problem set. There is no detailed solution for these questions.

Completing the review questions should take approximately 20 minutes.

*Standards Addressed (Common Core and AP):*

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is adapted from works attributed to David Lane and CK-12, which can be found here and here.

**4.2 Probability Distribution for a Discrete Random Variable**

*You just learned about the difference between discrete and continuous random variables, and now Subunit 4.2 focuses solely on discrete random variables, such as the outcome of a series of coin tosses or the number of boys in a family with four children.*

**Explanation**: Saylor Academy's Flexbook: Jill Schmidlkofer's *Advanced Probability and Statistics*: "Section 4.2: Probability Distribution for a Discrete Random Variable”; partially adapted from David Lane's *Online Statistics Education: A Multimedia Course of Study* and CK-12: *Advanced Probability and Statistics* (PDF)

Instructions: Read and take notes on Section 4.2. A probability distribution is an easy and condensed way of summarizing probabilities.

Reading this section and taking notes should take approximately 15 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.1
- CCSS.Math.Content.HSS-MD.A.3
- CCSS.Math.Content.HSS-MD.A.4
- CCSS.ELA-Literacy.RST.11-12.2

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is adapted from works attributed to David Lane and CK-12, which can be found here and here.

**Web Media**: YouTube: Khan Academy's "Introduction to Discrete Probability Distributions” (YouTube)

Instructions: Watch the video. A coin toss with unequal probabilities for heads and tails leads to a probability distribution where each probability is not equal to 0.5.

Watching this lecture should take approximately 5 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.1
- CCSS.Math.Content.HSS-MD.A.4
- AP III.A.4
- AP III.A.5

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is attributed to Khan Academy.

**Web Media:** SOPHIA: Ryan Backman's "Probability Distribution” (HTML5)

Instructions: Watch the video, which provides an overview of both discrete and continuous probability distributions. The discrete probability distribution is for all possible sums when you roll two dice.

Watching this lecture should take approximately 10 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.1
- CCSS.Math.Content.HSS-MD.A.4
- AP III.A.4
- AP III.A.5

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported.

**Checkpoint: **Saylor Academy's Flexbook: Jill Schmidlkofer's *Advanced Probability and Statistics*: "Section 4.2: Probability Distribution for a Discrete Random Variable”; partially adapted from David Lane's *Online Statistics Education: A Multimedia Course of Study* and CK-12: *Advanced Probability and Statistics* (PDF)

Instructions: Work on the review questions at the end of Section 4.2. When you create the probability distribution for each question, keep in mind that it is the value of the random variable X that "drives" the calculation of the probabilities. A short answer key is provided at the end of the problem set. For a detailed solution, click here.

Completing the review questions should take approximately 1 hour.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.1
- CCSS.Math.Content.HSS-MD.A.3
- CCSS.Math.Content.HSS-MD.A.4
- CCSS.ELA-Literacy.RST.11-12.2

**4.3 Mean and Standard Deviation of Discrete Random Variables**

*Just as we need to calculate the mean and standard deviation of raw data, we also need to do the same for data that is summarized in a probability distribution. Take heed: the term "expected value” is the same thing as the mean! Don't be misled or confused by the vocabulary. The symbol for the expected value is E(X), but remember that it is the same thing as the mean. *

**Explanation**: Saylor Academy's Flexbook: Jill Schmidlkofer's *Advanced Probability and Statistics*: "Section 4.3: Mean and Standard Deviation of Discrete Random Variables”; partially adapted from David Lane's *Online Statistics Education: A Multimedia Course of Study* and CK-12: *Advanced Probability and Statistics* (PDF)

Instructions: Read Section 4.3. Two new formulas are introduced in this section, for the mean and for the standard deviation of a probability distribution. While it is not necessary to memorize them, you should be able to use them for simple calculations. As you take notes, be sure you can follow each step in the examples.

Reading this section and taking notes should take approximately 50 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.1
- CCSS.Math.Content.HSS-MD.A.2
- CCSS.Math.Content.HSS-MD.A.3
- CCSS.Math.Content.HSS-MD.A.4
- CCSS.ELA-Literacy.RST.11-12.3
- AP III.A.4
- AP III.A.6

**Web Media**: Khan Academy's "Expected Value: E(X)” (YouTube)

Instructions: Watch the video. The expected value of a random variable is introduced in this video. The presenter gets a little off-topic at times, but he refocuses.

Watching this lecture should take approximately 15 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.2
- CCSS.Math.Content.HSS-MD.A.3
- AP III.A.4
- AP III.A.5
- AP III.A.6

**Did I Get This? Activity**: Khan Academy's "Expected Value” (HTML)

Instructions: Click on the link above and answer questions until you feel that you have mastered the calculation of the expected value. Remember, you can ask for a "hint” if you get stuck on any of the questions. Enter your answer in the answer box. You will receive feedback about your answer, and the entire solution will be presented for your review.

Completing this activity should take approximately 15 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.2
- CCSS.Math.Content.HSS-MD.A.3
- AP III.A.4
- AP III.A.5
- AP III.A.6

**Web Media**: SOPHIA: Daniel Nelson's "Mean and Variance of Probability Distributions” (HTML5)

Instructions: Watch the video, which provides a straightforward example of the calculation of the mean (expected value) and an explanation of the standard deviation of a discrete random variable about candy bar sales. Take notes and solve the problem as you watch the video. The presenter goes rather quickly, so you might have to stop the video to catch up with him. After the video is finished, try the example problem that is located just below the video screen.

Watching this lecture and taking notes should take approximately 10 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.2
- CCSS.Math.Content.HSS-MD.A.3
- AP III.A.4
- AP III.A.5
- AP III.A.6

Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 United States.

**Checkpoint**: Saylor Academy's Flexbook: Jill Schmidlkofer's *Advanced Probability and Statistics*: "Section 4.3: Mean and Standard Deviation of Discrete Random Variables”; partially adapted from David Lane's *Online Statistics Education: A Multimedia Course of Study* and CK-12: *Advanced Probability and Statistics* (PDF)

Instructions: Work on the review questions at the end of Section 4.3. Use the notes you took while reading the text to assist you in calculating the mean and variance of the probability distributions. A short answer key is provided at the end of the problem set. For a detailed solution, click here.

Completing the review questions should take approximately 1 hour.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.1
- CCSS.Math.Content.HSS-MD.A.2
- CCSS.Math.Content.HSS-MD.A.3
- CCSS.Math.Content.HSS-MD.A.4
- CCSS.ELA-Literacy.RST.11-12.3
- AP III.A.4
- AP III.A.6

**4.4 Sums and Differences of Independent Random Variables**

*Students often find this topic confusing and difficult to learn and apply. Keep focused! There are three main concepts to differentiate in this lesson:*

*Adding a*constant*to each value of a distribution, such as a teacher adding 10 points to every student's exam score.**Multiplying every value in a distribution by a*constant*, such as the teacher doubling every student's test score.**Adding two*distributions*together**, such as adding the weight of a student's sandwich to the weight of his or her beverage and creating a new distribution, the weight of the student's entire lunch.*

**Explanation**: Saylor Academy's Flexbook: Jill Schmidlkofer's *Advanced Probability and Statistics*: "Section 4.4: Sums and Differences of Independent Random Variables”; partially adapted from David Lane's *Online Statistics Education: A Multimedia Course of Study* and CK-12: *Advanced Probability and Statistics* (PDF)

Instructions: Read Section 4.4. The first part of this lesson teaches you how to create a probability distribution if you are given general information about an event's probabilities. The second part of the lesson teaches you how to combine two probability distributions. The formulas are a bit confusing, but they are broken down step by step. You should definitely take notes as you read, making sure that you understand each step of each sample problem's solution.

Reading this section and taking notes should take approximately 1 hour.

*Standards Addressed (Common Core and AP):*

- CCSS.ELA-Literacy.RST.11-12.3
- AP III.A.4
- AP III.A.6
- AP III.B.2

**Web Media**: SOPHIA: Emily LeBlanc-Perrone's "Probability and Random Variables - Part 3: Simulation & Transforming and Combining Random Variables” (HTML5)

Instructions: Click on the link above and watch the first video. This topic is often very difficult for students. Reread the text before watching this video, and note the difference between adding a constant to all values in a distribution versus adding one distribution to another distribution. After you finish the video, take the three-question quiz on the right side of the screen.

Watching this lecture and taking the quiz should take approximately 10 minutes.

*Standards Addressed (Common Core and AP):*

- AP III.A.6
- AP III.B.2

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported.

**Checkpoint:** Saylor Academy's Flexbook: Jill Schmidlkofer's *Advanced Probability and Statistics*: "Section 4.4: Sums and Differences of Independent Random Variables”; partially adapted from David Lane's *Online Statistics Education: A Multimedia Course of Study* and CK-12: *Advanced Probability and Statistics* (PDF)

Instructions: Work on the review questions at the end of Section 4.4. Use the example problems from the text to help you create the probability distributions in Problem 1 and Problem 3. Use the three rules for combining distributions for Problem 4. A short answer key is provided at the end of the problem set. For a detailed solution, click here.

Completing the review questions should take approximately 1 hour.

*Standards Addressed (Common Core and AP):*

- CCSS.ELA-Literacy.RST.11-12.3
- AP III.A.4
- AP III.A.6
- AP III.B.2

**4.5 The Binomial Distribution**

*The binomial distribution is a very important distribution in statistics. Read the text carefully, watch each of the videos, and perform the applets. We will see the binomial distribution later in this course, and mastery of it now will make later lessons much easier to grasp.*

**Explanation**: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 4.5: The Binomial Probability Distribution”; partially adapted from David Lane's Online Statistics Education: A Multimedia Course of Study and CK-12: Advanced Probability and Statistics (PDF)

Instructions: Read Section 4.5. Be sure that you can calculate binomial probabilities using both the binomial probability formula and your graphing calculator.

Reading this section should take approximately 1 hour and 30 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.1
- CCSS.Math.Content.HSS-MD.A.2
- CCSS.Math.Content.HSS-MD.A.3
- CCSS.Math.Content.HSS-MD.A.4
- AP III.A.4
- AP III.A.6

**Web Media**: Khan Academy's "Binomial Distribution 1” (YouTube)

Instructions: Watch the video. Make sure you have read the text before viewing this video and that you are familiar with the binomial probability formula. The presenter will show you how the formula was developed in an informal way.

Watching this lecture should take approximately 15 minutes.

*Standards Addressed (Common Core and AP):*

- AP III.A.4
- AP III.B.2

**Web Media**: Khan Academy's "Binomial Distribution 2” (YouTube)

Instructions: Watch the video, which is a continuation of the previous lesson.

Watching this lecture should take approximately 15 minutes.

*Standards Addressed (Common Core and AP):*

- AP III.A.4

**Web Media**: Khan Academy's "Binomial Distribution 3” (YouTube)

Instructions: Watch the video. The presenter uses a basketball example to further illustrate the use of the binomial distribution for calculating probabilities. Note that he uses the letter "k” instead of "x” in his formula, but the meaning and use of the two letters are identical.

Watching this lecture should take approximately 15 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.B.7
- AP III.A.4

**Web Media**: Khan Academy's "Binomial Distribution 4” (YouTube)

Instructions: Watch the video. The basketball example is continued. The presenter uses Microsoft Excel to simplify the calculations, and he creates a graph of the probability distribution.

Watching this lecture should take approximately 15 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.B.7
- AP III.A.4

**Web Media**: Khan Academy's "Expected Value of Binomial Distribution” (YouTube)

Instructions: Watch only the first 5:40 minutes of this video. If you would like to see how the formula for the expected value was derived, feel free to watch the entire video.

Watching this lecture should take approximately 5 minutes.

*Standards Addressed (Common Core and AP):*

**Web Media**: Prezi: Steve Mays's "4.2 Binomial Distributions” (Flash)

Instructions: Click on the link above, which takes you to a series of slides and videos. Feel free to view all of the material, but the presentation to focus on is the one shown as "Example 3,” entitled "Mean, Variance, & Standard Deviation of a Binomial Probability Distribution.” To go to this particular presentation, find the right arrow button on the black bar at the bottom of the video viewing area. Click it repeatedly to skip to the presentation for "Example 3.”

Watching this presentation should take approximately 5 minutes.

*Standards Addressed (Common Core and AP):*

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Activity**: Ron Blond's "The Binomial Distribution Applet” (Java)

Instructions: Have you ever guessed on a true/false test? You will get practice here. Follow the instructions carefully. Your first exercise will simulate a single individual guessing on a 20-question true-false test. You will simulate that person taking the test many times, and you will find that the student rarely passes the test.

In the second example, you will be able to see what happens if a group of 20 people take the test and all of them guess. The group scores will post to the graph.

You can change the probability from p = .5 (for true-false) to p = .25 (multiple-choice) test. Your hopes of passing diminish substantially!

Working on this activity should take approximately 15 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.B.5a
- CCSS.Math.Content.HSS-MD.B.5b
- CCSS.Math.Content.HSS-MD.B.6
- CCSS.Math.Content.HSS-MD.B.7
- AP III.A.4

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Web Media**: Penn State University's Eberly College of Science: David Little's "Plinko and the Binomial Distribution” (Java)

Instructions: Relax a little and let the little balls plink down the pegs of this simulated machine, technically called a quincunx. The idea is that when *p *= .5, each ball has a 50-50 chance of going right or left as it hits each peg. Can you guess where most of the balls will end up? You can see the shape of the graph that is made when many balls have completed their journey. What is the mean of this distribution? The variance?

Change p to a small value, such as .10. The balls now have a 90-10 chance of going to the left as they hit each peg. What happens to the paths of the balls now? What will be the shape of the graph that results after several hundred balls complete their journey?

Working with this applet should take approximately 20 minutes.

*Standards Addressed (Common Core and AP):*

- AP III.A.5

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Checkpoint**: Saylor Academy's Flexbook: Jill Schmidlkofer's *Advanced Probability and Statistics*: "Section 4.5: The Binomial Probability Distribution”; partially adapted from David Lane's *Online Statistics Education: A Multimedia Course of Study* and CK-12: *Advanced Probability and Statistics* (PDF)

Instructions: Work on the review questions at the end of Section 4.5. Use your calculator or the binomial probability formula to answer the questions. If you use your calculator, note the difference between Options A and B on the calculator. Option A is used to calculate a probability for a single outcome, whereas Option B is used for calculating cumulative probabilities (for multiple outcomes). Make sure that you use the binomial probability formula for at least one of the calculations. A short answer key is provided at the end of the problem set. For a detailed solution, click here.

Completing the review questions should take approximately 1 hour and 30 minutes.

*Standards Addressed (Common Core and AP):*

- CCSS.Math.Content.HSS-MD.A.1
- CCSS.Math.Content.HSS-MD.A.2
- CCSS.Math.Content.HSS-MD.A.3
- CCSS.Math.Content.HSS-MD.A.4
- AP III.A.4
- AP III.A.6