Unit 3 Activities

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Completing this unit should take approximately 11 hours and 15 minutes.

• Subunit 3.1: 1 hour
• Subunit 3.2: 1 hour
• Subunit 3.3: 1 hour
• Subunit 3.4: 3 hours
• Subunit 3.5: 4 hours
• Subunit 3.6: 2 hours and 15 minutes

Learning Outcomes

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

• List simple events and sample spaces.
• Know the symbols and operations of unions and intersections of sets.
• Know and use the Complement Rule to calculate the probability of an event.
• Calculate probabilities using the Addition Rule for mutually exclusive and non-mutually exclusive events.
• Calculate probabilities using the Multiplication Rule for independent and non-independent events.
• Calculate combinations and permutations.
• Use two-way tables as sample spaces for calculating joint, marginal, and conditional probabilities.
• Use probabilities to analyze real-world problems and make decisions.

Standards Addressed (Common Core and AP):

3.1 Events, Sample Spaces, and Probability

We begin our study of probability with the basics. We are introduced to new vocabulary and a statistical approach to describing outcomes of experiments. Understanding these basic ideas is critical for mastering the more challenging applications in the rest of this unit.

Explanation: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.1: Events, Sample Spaces, and Probability”; 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 3.1. Take notes on new vocabulary and work each example as it is presented. Notice the logical way in which the simple outcomes comprising a sample space are listed.

Reading this section and taking notes should take approximately 25 minutes.

Standards Addressed (Common Core and AP):

Instructions: Watch the video. This is a basic introduction to the vocabulary and calculation of basic probability.

Watching this lecture should take approximately 10 minutes.

Standards Addressed (Common Core and AP):

Checkpoint: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.2: Compound Events”: Compound Events; 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 3.1. Make sure you can create complete sample spaces. Review the concepts of sampling with replacement and sampling without replacement before attempting the review questions. 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 25 minutes.

Standards Addressed (Common Core and AP):

3.2 Compound Events: Concepts and Notation

This subunit continues vocabulary building by introducing you to compound events, which are composed of two or more simple events. We also learn about combining two or more sets by considering the union of the sets or the intersection of the sets.

Web Media: SOPHIA: Ms. Hess's "Finding the Union of Sets” (HTML5)

Instructions: Watch the video. The presenter gives excellent examples for finding the union of two sets. Pay close attention to her use of the "union” symbol U and to her use of brackets {  } when she expresses her answers.

Viewing this lecture should take approximately 5 minutes.

Standards Addressed (Common Core and AP):

Explanation: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.2: Compound Events”; 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 3.2. The concepts of union and intersection can be confusing, so work each example as you read the text. Taking notes will help you keep the concepts of union and intersection differentiated.

Reading this section and taking notes should take approximately 30 minutes.

Standards Addressed (Common Core and AP):

Web Media: SOPHIA: Ms. Hess's "Finding the Intersection of Sets” (HTML5)

Instructions: Watch the video. The presenter gives excellent examples for finding the intersection of two sets. Pay close attention to her use of the "intersection” symbol  and to her use of brackets {  } when she expresses her answers.

Viewing this lecture should take approximately 5 minutes.

Standards Addressed (Common Core and AP):

Checkpoint: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.2: Compound Events”; 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 question at the end of Section 3.2. Pay close attention to the question asked - either intersection or union. 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 20 minutes.

Standards Addressed (Common Core and AP):

3.3 The Complement of an Event

If the weather reporter says there is a 30% chance of rain today, then there is a 70% chance that it will not rain. These two events, "rain” and "not rain,” are complements. This straightforward concept is used often in statistics, so it is wise to master it.

Explanation: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.3: The Complement of an Event”; 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 3.3. Take notes as you read. Note the notation for the complement of event A, which is A' (pronounced "A prime” or "A complement”). The Venn diagram is useful for understanding the complement of an event and for calculating probabilities.

Reading this section and taking notes should take approximately 25 minutes.

Standards Addressed (Common Core and AP):

Web Media: SOPHIA: Ryan Backman's "Complement of an Event” (HTML5)

Instructions: Watch the video. The presenter gives excellent examples for finding the complement of an event. Pay close attention to his use of the symbol A' (pronounced "A prime” or "A complement") for denoting the complement of an event.

Watching this lecture should take approximately 5 minutes.

Standards Addressed (Common Core and AP):

Checkpoint: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.3: The Complement of an Event”; 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 3.3. Use your notes from Section 3.2 and Section 3.3 to assist you in answering the questions. 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 30 minutes.

Standards Addressed (Common Core and AP):

3.4 Conditional Probability

Conditional probability is one of the most challenging concepts in statistics. It takes time to master it. Please read the text material very carefully and slowly, making sure you are actively reading for comprehension. Watch the videos, complete the checkpoints, and come back to the text and reread it.

Explanation: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.4: Conditional Probability”; 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 3.4. Read it carefully; this is a challenging section. Take notes, and follow every example one step at a time.

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

Standards Addressed (Common Core and AP):

Reading: Florida Gulf Coast University: Thomas Harrington's "Simple, Joint, Marginal and Conditional Probabilities” as adapted for use by Saylor Academy (PDF)

Instructions: Read the material carefully. It is important to be able to interpret data summarized in a table. You will calculate simple probabilities, marginal probabilities, joint probabilities, and conditional probabilities in this reading.

Reading and summarizing this material should take approximately 45 minutes.

Standards Addressed (Common Core and AP):

Instructions: Watch the video. The concept of conditional (dependent) probability is introduced here in a straightforward way. Pay close attention to the video, especially the notation used for conditional probability P(A|B), which is verbalized as "the conditional probability of A, given that B occurred.”

Watching this lecture should take approximately 10 minutes.

Standards Addressed (Common Core and AP):

Instructions: Watch the video. The presenter gives you an intuitive and simple mathematical approach to optimal strategy for this popular television show using conditional probabilities.

Watching this lecture should take approximately 10 minutes.

Standards Addressed (Common Core and AP):

Interactive Lab: Shodor: "Simple Monty Hall” (Java)

Instructions: This is an applet that gives you some play time as a contestant on the popular TV show "Let's Make a Deal.” You are now the contestant and you make the decision about whether you should keep the door you originally chose or whether you should switch.

Try 100 simulations. Your wins and losses will be displayed. At the end of 100 simulations, you will be a believer that the presenter was absolutely correct in the Khan Academy video about the same topic.

Note that you will click once for your first choice. Then one of the doors you did NOT choose will display its nonwinning prize. After this, you will have two choices: You will either click again on your original door, or you will switch by clicking on the door you did not originally choose. The action is very fast, and it will probably take you a few times to become familiar with how the game works. Just work slowly at first, and you will get into the flow of the game.

Performing this exercise should take approximately 15 minutes.

Standards Addressed (Common Core and AP):

Instructions: Click on the link above to watch the video. This is a more advanced treatment of conditional probability, in which the probability of flipping a certain number of heads in a row isdependent on which coin you choose from the bag, and you don't know ahead of time which coin it will be.

Watching this lecture should take approximately 10 minutes.

Standards Addressed (Common Core and AP):

Checkpoint: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.4: Conditional Probability”; 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 3.4. Try to answer each question using both the formula for conditional probability and the natural frequencies approach. You will discover that some problem types are easier to solve by using the formula, and others are easier to solve by using the natural frequencies approach. 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 40 minutes.

Standards Addressed (Common Core and AP):

3.5 Probability Rules: Addition and Multiplication Rules

Another tough probability concept is the use of the addition rule and the multiplication rule. Go through the text material slowly and carefully. Pay special attention to the difference between mutually exclusive events and independent events. View the videos carefully; they will help to clarify the concepts.

Explanation: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.5: Addition and Multiplication Rules”; 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 3.5. Carefully take notes and follow each example step-by-step. Make sure you know the difference between mutually exclusive events and independent events.

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

Standards Addressed (Common Core and AP):

Web Media: Khan Academy's "Probability with Playing Cards and Venn Diagrams” (YouTube)

Instructions: Watch the video. The presenter introduces us to the addition rule by using a deck of cards and by illustrating his work with a Venn diagram.

Watching this lecture should take approximately 15 minutes.

Standards Addressed (Common Core and AP):

Instructions: Watch the video. The presenter gives us an intuitive introduction to the addition rule by considering a bag filled with boxes and cubes of different colors. He uses a Venn diagram to "map” the experiment.

Watching this lecture should take approximately 15 minutes.

Standards Addressed (Common Core and AP):

Instructions: Watch the video. The presenter introduces the multiplication rule by listing all possible outcomes in the sample space and then showing how the multiplication rule makes practical sense.

Watching this lecture should take approximately 10 minutes.

Standards Addressed (Common Core and AP):

Instructions: Watch the video. The presenter creates the entire sample space for the rolls of two dice. He then shows how to calculate the probability of getting doubles, based first on the sample space, and then by using the multiplication rule.

Watching this lecture should take approximately 10 minutes.

Standards Addressed (Common Core and AP):

Web Media: Khan Academy's "Example: Getting Two Questions Right on an Exam” (YouTube)

Instructions: Watch the video. This video presents an everyday example that uses the multiplication rule for independent events for its solution.

Watching this lecture should take approximately 5 minutes.

Standards Addressed (Common Core and AP):

Web Media: Khan Academy's "Example: Is an Event Independent or Dependent?” (YouTube)

Instructions: Watch the video. This raffle example is similar to drawing marbles from a bag. The presenter shows us how to intuitively decide if two events are independent or not independent.

Watching this lecture should take approximately 5 minutes.

Standards Addressed (Common Core and AP):

Checkpoint: Khan Academy's "Independent Probability” (HTML)

Instructions: Click on the link above and answer as many questions as you need until you feel confident about independent probabilities. Enter your answer in the box at the right of the screen, and press the "Check Answer” box. You can see a detailed solution by clicking on the "Show Solution” button.

Completing this task should take approximately 30 minutes.

Checkpoint: Khan Academy's "Dependent Probability” (HTML)

Instructions: Click on the link above and answer as many questions as you need until you feel confident about dependent probabilities. Enter your answer in the box at the right of the screen, and press the "Check Answer” box. You can see a detailed solution by clicking on the "Show Solution” button.

Completing this task should take approximately 30 minutes.

Checkpoint: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.5: Addition and Multiplication Rules”; 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 3.5. For each question, consider the sample space. Also ask yourself if the events in question are mutually exclusive or independent. Remember that you will use the addition rule if the question includes the "union” symbol or if it asks an "or” question. You will use the multiplication rule if the question includes the "intersection” symbol or if it asks an "and” question. 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):

3.6 Basic Counting Rules

In this lesson, concentrate your attention on the difference between permutations and combinations. Study the rules and formulas of combinations carefully, because combinations will appear again when you study the binomial distribution in a later unit.

Additional Note: This is the final section of Unit 3. If you scroll down below the review question set at the end of the section in the text, you will see a list of important ideas and formulas from the entire unit. This is an extremely helpful list for review.

Explanation: Saylor Academy's Flexbook: Jill Schmidlkofer's Advanced Probability and Statistics: "Section 3.6: Basic Counting Rules”; 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 3.6. Take notes first on the use of the Multiplicative Rule for Counting, such as the number of different outfits you can make if you have five tops and three bottoms. Then work on the Rule for Permutations, which is used when the order of items is important, such as the number of ways you can put five books into a bookshelf. Finally, give attention to the Rule for Combinations, which is used when the order of items is not important, such as choosing three people to plan a party from a club with 40 members.

Reading this section and taking notes should take approximately 40 minutes.

Standards Addressed (Common Core and AP):

Instructions: Watch the video. The key concept in permutations is that the order in which events happen is important. The presenter introduces us to the general formula for calculating the number of permutations that can occur for a set of events.

Don't let the mathematical symbol for a factorial scare you. Remember that 5 factorial, for example, is symbolically expressed as 5! and it is equal to 5×4×3×2×1=120.

Watching this lecture should take approximately 15 minutes.

Standards Addressed (Common Core and AP):

Instructions: Watch the video. When we study combinations, the order of the events doesn't matter. That is what differentiates combinations from permutations. The presenter concentrates on combinations in this video, so listen carefully when he talks about the fact that the order doesn't matter when we do combinatorial math.

Watching this lecture should take approximately 15 minutes.

Standards Addressed (Common Core and AP):

Instructions: Watch the video. This is an excellent example of the application of permutations.

Watching this lecture should take approximately 5 minutes.

Standards Addressed (Common Core and AP):

Checkpoint: Khan Academy's "Permutations and Combinations” (HTML)

Instructions: Click on the link above and answer as many questions as you need until you feel confident about when to calculate a permutation and when to calculate a combination and which formula to use to obtain a correct answer. Enter each answer in the box at the right of the screen, and press the "Check Answer” box for immediate feedback. You can get hints, or you can obtain a detailed answer to each problem by clicking on the boxes on the right side of the screen.

Note: If you have difficulty with this assessment, look at the left side of the assessment screen. At the bottom, you will see two starred items; one is called Combinations and the other is called Permutations. These are problem sets for each individual topic. Click on one of these, and you will get practice for each topic individually. Later, you can return to the assessment above, where the two types of problems are mixed together.

Completing this task should take approximately 35 minutes.

Standards Addressed (Common Core and AP):