K12MATH010: Geometry

Unit 6: Applications of Probability   Ever played darts? Geometric probability helps us understand the likelihood of hitting the bull’s-eye. Ever taken a train or a bus to meet a friend? Geometric probability helps us understand your likelihood of arriving on time. This course ends by revisiting probability, building on concepts learned in previous grades but now using geometric probability models where possible. You will use set theory to compute and interpret theoretical and experimental probabilities of compound events, examining mutually exclusive events, independent events, and conditional probability. You will then apply this knowledge to make informed decisions and evaluation decisions made using probability.[[6]](#_ftn1)

[6]Ibid

Completing this unit should take you approximately 12 hours, 30 minutes.

☐    Subunit 6.1: 4 hours, 30 minutes

☐    Subunit 6.2: 8 hours

Unit6 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- Identify and explain independence and conditional probability. - Use independence and conditional probability to interpret data using two-way tables. - Apply the rules of probability to compute the probabilities of compound events. - Evaluate the outcomes of decisions using probability.

6.1 Independent and Dependent Events   In this subunit, you will review the basic concepts of probability and how they relate to your everyday life. You will examine the probability of independent events taking place—earning an A in Spanish class and earning a B in Biology, for example. You will also examine the probability of dependent events—having a day off and going to an art museum, for example. The concepts of mutually exclusive and mutually inclusive events are related to this. When determining the probability of several events, these terms are important to understand in order to avoid double counting. The resources in this subunit provide explanatory content as well as practice problems that allow you to receive immediate feedback on your progress and understanding.

6.1.1 Venn Diagrams   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Venn Diagrams” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Venn Diagrams” (HTML)

Instructions: Please click on the link above, and watch the three videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon reviewing this section, you should be able to read Venn diagrams and use them to demonstrate the various possible outcomes of an experiment.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.A.1](http://www.corestandards.org/Math/Content/HSS/CP/A/1)
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[CCSS.Math.Content.HSS-CP.A.5](http://www.corestandards.org/Math/Content/HSS/CP/A/5)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
``````

Instructions: Please click on the link above, watch the video, and take notes, following along with the examples the presenter provides. This video examines the use of Venn diagrams with playing cards.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSS-CP.A.1
●      CCSS.Math.Content.HSS-CP.A.5

Instructions: Please click on the link above. This page provides a series of practice problems that allow you to calculate basic probability in a variety of situations. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSS-CP.A.1

6.1.2 Independent Events and Sample Spaces   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Independent Events and Sample Spaces” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Independent Events and Sample Spaces” (HTML)

Instructions: Please click on the link above, and watch the three videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon reading this section, you should be able to identify independent events and calculate the probability of an outcome of independent events taking place.

Completing this activity should take approximately 30 minutes.

`````` Standards Addressed (*Common Core*):

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[CCSS.Math.Content.HSS-CP.A.1](http://www.corestandards.org/Math/Content/HSS/CP/A/1)
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[CCSS.Math.Content.HSS-CP.A.2](http://www.corestandards.org/Math/Content/HSS/CP/A/2)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
``````

Instructions: Please click on the link above. This page provides a series of practice problems that allow you to calculate the probability independent events. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSS-CP.A.1
●      CCSS.Math.Content.HSS-CP.A.2

6.1.3 Dependent Events and Sample Spaces   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Dependent Events and Sample Spaces” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Dependent Events and Sample Spaces” (HTML)

Instructions: Please click on the link above, and watch the three videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon reading this section, you should be able to identify dependent events and calculate the probability of an outcome involving such events.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.A.1](http://www.corestandards.org/Math/Content/HSS/CP/A/1)
●
[CCSS.Math.Content.HSS-CP.A.2](http://www.corestandards.org/Math/Content/HSS/CP/A/2)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
``````

Instructions: Please click on the link above. This page provides a series of practice problems that allow you to calculate the probability of dependent events. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSS-CP.A.1
●      CCSS.Math.Content.HSS-CP.A.2

6.1.4 Mutually Exclusive Events   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Mutually Exclusive Events” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Mutually Exclusive Events” (HTML)

Instructions: Please click on the link above, and watch the two videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon reading this section, you should be able to identify mutually exclusive events and calculate the probability of outcomes in such events.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.A.1](http://www.corestandards.org/Math/Content/HSS/CP/A/1)
●
[CCSS.Math.Content.HSS-CP.A.2](http://www.corestandards.org/Math/Content/HSS/CP/A/2)
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[CCSS.Math.Content.HSS-CP.A.4](http://www.corestandards.org/Math/Content/HSS/CP/A/4)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
``````

6.1.5 Mutually Inclusive Events   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Mutually Inclusive Events” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Mutually Inclusive Events” (HTML)

Instructions: Please click on the link above, and watch the three videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon reading this section, you should be able to identify mutually inclusive events and calculate the probability of outcomes in such events.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.A.1](http://www.corestandards.org/Math/Content/HSS/CP/A/1)
●
[CCSS.Math.Content.HSS-CP.A.2](http://www.corestandards.org/Math/Content/HSS/CP/A/2)
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[CCSS.Math.Content.HSS-CP.B.7](http://www.corestandards.org/Math/Content/HSS/CP/B/7)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
``````

6.2 Conditional Probability   This subunit builds on the previous section. You will examine permutations and combinations, looking at how order may affect the probability (as with permutations) or may not affect the probability (as with combinations). You can use permutations to examine the probability of correctly guessing your locker combination or to decide how many different groups of 4 your teacher can arrange from your class of 24 students. Additionally, you will examine conditional probability, which exists when the probability of the second event depends on the outcome of the first event. The resources in this subunit provide explanatory content as well as practice problems that allow you to receive immediate feedback on your progress and understanding.

6.2.1 Tree Diagrams   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Tree Diagrams” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Tree Diagrams” (HTML)

Instructions: Please click on the link above, and watch the three videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon completing this section, you should be able to read and create tree diagrams to illustrate the possible outcomes of an event.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.A.5](http://www.corestandards.org/Math/Content/HSS/CP/A/5)
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[CCSS.Math.Content.HSS-CP.B.7](http://www.corestandards.org/Math/Content/HSS/CP/B/7)
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[CCSS.Math.Content.HSS-CP.B.8](http://www.corestandards.org/Math/Content/HSS/CP/B/8)

displayed on the webpage above.
``````

6.2.2 Permutations and Combinations Compared   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Permutations and Combinations Compared” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Permutations and Combinations Compared” (HTML)

Instructions: Please click on the link above, and watch the three videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon reading this section, you should be able to identify permutations and use them to determine the total number of arrangements possible in an ordered set of objects.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.B.6](http://www.corestandards.org/Math/Content/HSS/CP/B/6)
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[CCSS.Math.Content.HSS-CP.B.9](http://www.corestandards.org/Math/Content/HSS/CP/B/9)

displayed on the webpage above.
``````
• Did I Get This? Activity: Khan Academy’s “Permutations and Combinations” Link: Khan Academy’s “Permutations and Combinations” (HTML)

Instructions: Please click on the link above. This page provides a series of practice problems that allow you to calculate the probability of events involving permutations and combinations. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSS-CP.B.6
●      CCSS.Math.Content.HSS-CP.B.9

6.2.3 Permutation Problems   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Permutation Problems” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Permutation Problems” (HTML)

Instructions: Please click on the link above, and watch the two videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice are optional, as this course provides you with practice from other sources. Upon reading this section, you should be able to solve problems requiring the use of permutations.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.B.6](http://www.corestandards.org/Math/Content/HSS/CP/B/6)
●
[CCSS.Math.Content.HSS-CP.B.9](http://www.corestandards.org/Math/Content/HSS/CP/B/9)

displayed on the webpage above.
``````

Instructions: Please click on the link above. This page provides a series of practice problems that allow you to calculate the probability of events involving permutations. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSS-CP.B.6
●      CCSS.Math.Content.HSS-CP.B.9

6.2.4 Permutations with Repetition   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Permutations with Repetition” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Permutations with Repetition” (HTML)

Instructions: Please click on the link above, and watch the two videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon reading this section, you should be able to identify permutations with repetition and calculate the probability of outcomes in such events.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.B.6](http://www.corestandards.org/Math/Content/HSS/CP/B/6)
●
[CCSS.Math.Content.HSS-CP.B.9](http://www.corestandards.org/Math/Content/HSS/CP/B/9)

displayed on the webpage above.
``````
• Checkpoint: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Permutation Problems Exercises” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Permutation Problems Exercises” (HTML)

Instructions: Please click on the link above, and then click on the tab labeled “Exercises.” This page will test your knowledge of permutations by providing you with a series of multiple-choice questions on the topic. Take your time and be sure you have scratch paper available to complete the calculations. Upon submitting each answer, the program will provide you with immediate feedback. For questions that you answer incorrectly, take a few minutes to understand your error and the correct answer.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSS-CP.B.6
●      CCSS.Math.Content.HSS-CP.B.9

6.2.5 Combinations   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Combinations” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Combinations” (HTML)

Instructions: Please click on the link above, and watch the three videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon reading this section, you should be able to identify combinations and calculate the probability of outcomes in such events.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.B.6](http://www.corestandards.org/Math/Content/HSS/CP/B/6)
●
[CCSS.Math.Content.HSS-CP.B.9](http://www.corestandards.org/Math/Content/HSS/CP/B/9)

displayed on the webpage above.
``````

6.2.6 Combination Problems   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Combination Problems” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Combination Problems” (HTML)

Instructions: Please click on the link above, and watch the two videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon reading this section, you should be able to solve problems involving combinations.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.B.6](http://www.corestandards.org/Math/Content/HSS/CP/B/6)
●
[CCSS.Math.Content.HSS-CP.B.9](http://www.corestandards.org/Math/Content/HSS/CP/B/9)

displayed on the webpage above.
``````

Instructions: Please click on the link above. This page provides a series of practice problems that allow you to calculate the probability of outcomes in events involving combinations. Each question has explanations worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed about six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSS-CP.B.6
●      CCSS.Math.Content.HSS-CP.B.9

• Checkpoint: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Combination Problems Exercises” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Combination Problems Exercises” (HTML)

Instructions: Please click on the link above, and then click on the tab labeled “Exercises.” This page will test your knowledge of combinations by providing you with a series of multiple-choice questions on the topic. Take your time and be sure you have scratch paper available to complete the calculations. Upon submitting each answer, the program will provide you with immediate feedback. For questions that you answer incorrectly, take a few minutes to understand your error and the correct answer.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSS-CP.B.6
●      CCSS.Math.Content.HSS-CP.B.9

6.2.7 Conditional Probability   - Explanation: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Conditional Probability” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Conditional Probability” (HTML)

Instructions: Please click on the link above, and watch the three videos provided. Next read the information that follows the videos, take notes, and complete the examples as you go. Answers follow each example. The guided practice and practice problems are optional, as this course provides you with practice from other sources. Upon reading this section, you should be able to identify conditional probability and calculate the probability of outcomes in such events.

Completing this activity should take approximately 30 minutes.

`````` ●
[CCSS.Math.Content.HSS-CP.B.6](http://www.corestandards.org/Math/Content/HSS/CP/B/6)
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[CCSS.Math.Content.HSS-CP.B.9](http://www.corestandards.org/Math/Content/HSS/CP/B/9)

displayed on the webpage above.
``````

Instructions: Please click on the link above, where you will see a page with four tabs. Click on the tab labeled “Learner,” and then download the worksheets titled “Advanced Monty Hall Exploration Questions” and “Monty Hall Tally Table.” Next, click on the tab labeled “Activity” to complete the activity. The “Help” tab provides further instructions on how the activity works if you have difficulties.

Completing this activity should take approximately 45 minutes.

●      CCSS.Math.Content.HSS-CP.A.5
●      CCSS.Math.Content.HSS-CP.B.6
●      CCSS.Math.Content.HSS-CP.B.9

• Checkpoint: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Conditional Probability Exercises” Link: CK-12: Basic Probability and Statistics Concepts—A Full Course: “Conditional Probability Exercises” (HTML)

Instructions: Please click on the link above, and then click on the tab labeled “Exercises.” This page will test your knowledge of conditional probability by providing you with a series of multiple-choice questions on the topic. Take your time and be sure you have scratch paper available to complete the calculations. Upon submitting each answer, the program will provide you with immediate feedback. For questions that you answer incorrectly, take a few minutes to understand your error and the correct answer.

Completing this activity should take approximately 30 minutes.