# Cluster 1: Understand independence and conditional probability and use them to interpret data. (Algebra 2 - Additional Cluster)

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.912.S-CP.1
Title: Understand independence and conditional probability and use them to interpret data. (Algebra 2 - Additional Cluster)
Type: Cluster
Subject: Mathematics
Domain-Subdomain: Statistics & Probability: Conditional Probability & the Rules of Probability

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.912.S-CP.1.AP.1a
Describe events as subsets of a sample space using characteristics or categories. For example: When rolling a die the sample space is 1, 2, 3, 4, 5, 6. The even numbers would be a subset of the sample space.
MAFS.912.S-CP.1.AP.1b
Describe the union of events in a sample space. For example: Event A contains soccer players, event B contains football players. The union of the sets is football players and soccer players all together.
MAFS.912.S-CP.1.AP.1c
Describe the intersection of events in a sample space. For example: Event A contains soccer players, event B contains football players. Intersection of the sets is players that participate in both soccer and football.
MAFS.912.S-CP.1.AP.1d
Describe the complement of events in a sample space. For example: Event A contains soccer players, event B contains football players. The complement of Event B is all players that are not football players.
MAFS.912.S-CP.1.AP.2a
Describe the characteristics that make events independent.
MAFS.912.S-CP.1.AP.2b
Calculate the probability of events A and B occurring together.

P(A and B) = P(A) × P(B)

MAFS.912.S-CP.1.AP.3a
Using a two-way table, find the conditional probability of A given B.
MAFS.912.S-CP.1.AP.3b
Identify when two events are independent. P(A and B) ÷ P(A) = P(B)
MAFS.912.S-CP.1.AP.4a
Select or make an appropriate statement based on a two-way frequency table.
MAFS.912.S-CP.1.AP.5a
Select or make an appropriate statement based on real-world examples of conditional probability.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Assessments

Sample 2 - High School Algebra 2 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 3 - High School Algebra 2 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 1 - High School Algebra 2 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

## Educational Software / Tool

This Excel spreadsheet allows the educator to input data into a two way frequency table and have the resulting relative frequency charts calculated automatically on the second sheet. This resource will assist the educator in checking student calculations on student-generated data quickly and easily.

Steps to add data: All data is input on the first spreadsheet; all tables are calculated on the second spreadsheet

2. Input joint frequency data.
3. Click the second tab at the bottom of the window to see the automatic calculations.

Type: Educational Software / Tool

## Lesson Plans

STEM Genetics Board Game:

This is a STEM challenge to assist in teaching the probability of traits being passed down from parents to offspring by creating and playing a board game.

Type: Lesson Plan

Comedy vs. Action Movies Frequency Interpretation:

Using a completed survey of male and female student interest in comedy vs. action movies, the students will create a two-way frequency table using actual data results, fraction results, and percent results. The students will then act as the movie producer and interpret the data to determine if it is in their best interest to make a comedy or action movie. As the Summative Assessment, the student will take on the job/role of an actor/actress and interpret the data to support their decision.

Type: Lesson Plan

Casino Royale:

This lesson will use different games of chance to understand the difference between dependent and independent conditional probability.

Type: Lesson Plan

Modeling Conditional Probabilities 1: Lucky Dip:

This lesson unit is intended to help you assess how well students are able to understand conditional probability, represent events as a subset of a sample space using tables and tree diagrams, and communicate their reasoning clearly.

Type: Lesson Plan

Modeling Conditional Probabilities 2:

This lesson unit is intended to help you assess how well students understand conditional probability, and, in particular, to help you identify and assist students who have the following difficulties representing events as a subset of a sample space using tables and tree diagrams and understanding when conditional probabilities are equal for particular and general situations.

Type: Lesson Plan

Medical Testing:

This lesson unit is intended to help you assess how well students are able to:

• make sense of a real life situation and decide what math to apply to the problem
• understand and calculate the conditional probability of an event A, given an event B, and interpret the answer in terms of a model
• represent events as a subset of a sample space using tables, tree diagrams, and Venn diagrams
• interpret the results and communicate their reasoning clearly

Type: Lesson Plan

Human Venn Diagram:

This activity is to strengthen students understanding of Venn diagrams, where the class becomes the problem. The class will be able to physically move and see how and why elements belong in each section of the Venn diagram.

Type: Lesson Plan

## Perspectives Video: Expert

Let's Make a Math Deal:

Should I keep my choice or switch?  Learn more about the origins and probability behind the Monty Hall door picking dilemma and how Game Theory and strategy effect the probability.

Type: Perspectives Video: Expert

Rain and Lightning:

This problem solving task challenges students to determine if two weather events are independent, and use that conclusion to find the probability of having similar weather events under certain conditions.

The task is intended to address sample space, independence, probability distributions and permutations/combinations.

Lucky Envelopes:

Cards and Independence:

This problem solving task lets students explore the concept of independence of events.

Breakfast Before School:

The purpose of this task is to assess a student's ability to explain the meaning of independence in a simple context.

How Do You Get to School?:

This task requires students to use information in a two-way table to calculate a probability and a conditional probability.

The Titanic 2:

This task lets students explore the concepts of probability as a fraction of outcomes and using two-way tables of data.

The Titanic 1:

This task asks students to calculate probabilities using information presented in a two-way frequency table.

The Titanic 3:

This problem solving task asks students to determine probabilities and draw conclusions about the survival rates on the Titanic by consulting a table of data.

This lesson is designed to introduce students to statistical situations where the probabilities or outcomes might not be what is first expected. The lesson provides links to discussions and activities motivated by the idea of unexpected answers. Finally, the lesson provides links to follow-up lessons designed for use in succession with an introduction to probability and unexpected answers in probability.

## Teaching Idea

Conditional Probability and Probability of Simultaneous Events:

This lesson is designed to further students' practice with probability as well as introduce them to conditional probability and probabilities of simultaneous independent events. The lesson provides links to discussions and activities related to conditional and simultaneous probabilities as well as suggested ways to integrate them into the lesson. Finally, this lesson provides links to follow-up lessons designed for use in succession with this one.

Type: Teaching Idea

## Text Resources

The Logic of Drug Testing:

This informational text resource is intended to support reading in the content area. This article explores the reliability of drug tests for athletes, using mathematics. The author attempts to address this issue by relating drug tests to conditional probability. Throughout the text, various numbers that affect the calculation of a reliable probability are discussed. Numbers such as test sensitivity, test specificity, and weight of evidence are related to Bayes' theorem, which is ultimately used to calculate the conditional probability.

Type: Text Resource

Understanding Uncertainty: What Was the Probability of Obama Winning?:

This informational text resource is intended to support reading in the content area. The article examines various factors that changed the uncertainty of whether Barack Obama would win the 2008 election. Specifically,the article discusses probability, the science of quantifying uncertainty. The article questions common methods for assessing probability where symmetrical outcomes are assumed. Finally, the author explains how to use past evidence to assess the chances of future events.

Type: Text Resource

## Video/Audio/Animation

MIT BLOSSOMS - Taking Walks, Delivering Mail: An Introduction to Graph Theory:

This learning video presents an introduction to graph theory through two fun, puzzle-like problems:"The Seven Bridges of Königsberg" and "The Chinese Postman Problem". Any high school student in a college-preparatory math class should be able to participate in this lesson. Materials needed include: pen and paper for the students; if possible, printed-out copies of the graphs and image that are used in the module; and a blackboard or equivalent. During this video lesson, students will learn graph theory by finding a route through a city/town/village without crossing the same path twice. They will also learn to determine the length of the shortest route that covers all the roads in a city/town/village. To achieve these two learning objectives, they will use nodes and arcs to create a graph and represent a real problem. This video lesson cannot be completed in one usual class period of approximately 55 minutes. It is suggested that the lesson be presented over two class sessions.

Type: Video/Audio/Animation

## Virtual Manipulatives

Spinner:

In this activity, students adjust how many sections there are on a fair spinner then run simulated trials on that spinner as a way to develop concepts of probability. A table next to the spinner displays the theoretical probability for each color section of the spinner and records the experimental probability from the spinning trials. This activity allows students to explore the topics of experimental and theoretical probability by seeing them displayed side by side for the spinner they have created. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Interactive Marbles:

This online manipulative allows the student to simulate placing marbles into a bag and finding the probability of pulling out certain combinations of marbles. This allows exploration of probabilities of multiple events as well as probability with and without replacement. The tabs above the applet provide access to supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

## Worksheet

Replacement and Probability:

This lesson is designed to develop students' understanding of sampling with and without replacement and its effects on the probability of drawing a desired object. The lesson provides links to discussions and activities related to replacement and probability as well as suggested ways to work them into the lesson. Finally, the lesson provides links to follow-up lessons that are designed to be used in succession with the current one.

Type: Worksheet

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Educational Software / Tool

This Excel spreadsheet allows the educator to input data into a two way frequency table and have the resulting relative frequency charts calculated automatically on the second sheet. This resource will assist the educator in checking student calculations on student-generated data quickly and easily.

Steps to add data: All data is input on the first spreadsheet; all tables are calculated on the second spreadsheet

2. Input joint frequency data.
3. Click the second tab at the bottom of the window to see the automatic calculations.

Type: Educational Software / Tool

Rain and Lightning:

This problem solving task challenges students to determine if two weather events are independent, and use that conclusion to find the probability of having similar weather events under certain conditions.

The task is intended to address sample space, independence, probability distributions and permutations/combinations.

Lucky Envelopes:

Cards and Independence:

This problem solving task lets students explore the concept of independence of events.

Breakfast Before School:

The purpose of this task is to assess a student's ability to explain the meaning of independence in a simple context.

The Titanic 2:

This task lets students explore the concepts of probability as a fraction of outcomes and using two-way tables of data.

The Titanic 1:

This task asks students to calculate probabilities using information presented in a two-way frequency table.

The Titanic 3:

This problem solving task asks students to determine probabilities and draw conclusions about the survival rates on the Titanic by consulting a table of data.

## Virtual Manipulatives

Spinner:

In this activity, students adjust how many sections there are on a fair spinner then run simulated trials on that spinner as a way to develop concepts of probability. A table next to the spinner displays the theoretical probability for each color section of the spinner and records the experimental probability from the spinning trials. This activity allows students to explore the topics of experimental and theoretical probability by seeing them displayed side by side for the spinner they have created. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Interactive Marbles:

This online manipulative allows the student to simulate placing marbles into a bag and finding the probability of pulling out certain combinations of marbles. This allows exploration of probabilities of multiple events as well as probability with and without replacement. The tabs above the applet provide access to supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

Rain and Lightning:

This problem solving task challenges students to determine if two weather events are independent, and use that conclusion to find the probability of having similar weather events under certain conditions.

The task is intended to address sample space, independence, probability distributions and permutations/combinations.

Lucky Envelopes:

Cards and Independence:

This problem solving task lets students explore the concept of independence of events.

Breakfast Before School:

The purpose of this task is to assess a student's ability to explain the meaning of independence in a simple context.

The Titanic 2:

This task lets students explore the concepts of probability as a fraction of outcomes and using two-way tables of data.