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Standard 4 : Use and interpret independence and probability.
Cluster Standards

This cluster includes the following benchmarks.

Visit the specific benchmark webpage to find related instructional resources.

  • MA.912.DP.4.1 : Describe events as subsets of a sample space using characteristics, or categories, of the outcomes, or as unions, intersections or complements of other events.
  • MA.912.DP.4.2 : Determine if events A and B are independent by calculating the product of their probabilities.
  • MA.912.DP.4.3 : Calculate the conditional probability of two events and interpret the result in terms of its context.
  • MA.912.DP.4.4 : Interpret the independence of two events using conditional probability.
  • MA.912.DP.4.5 : Given a two-way table containing data from a population, interpret the joint and marginal relative frequencies as empirical probabilities and the conditional relative frequencies as empirical conditional probabilities. Use those probabilities to determine whether characteristics in the population are approximately independent.
  • MA.912.DP.4.6 : Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
  • MA.912.DP.4.7 : Apply the addition rule for probability, taking into consideration whether the events are mutually exclusive, and interpret the result in terms of the model and its context.
  • MA.912.DP.4.8 : Apply the general multiplication rule for probability, taking into consideration whether the events are independent, and interpret the result in terms of the context.
  • MA.912.DP.4.9 : Apply the addition and multiplication rules for counting to solve mathematical and real-world problems, including problems involving probability.
  • MA.912.DP.4.10 : Given a mathematical or real-world situation, calculate the appropriate permutation or combination.
Cluster Information
Number:
MA.912.DP.4
Title:
Use and interpret independence and probability.
Type:
Standard
Subject:
Mathematics (B.E.S.T.)
Grade:
912
Strand
Data Analysis and Probability
Cluster Access Points

This cluster includes the following Access Points.

  • MA.912.DP.4.AP.1 : Given a sample space, select a subset of the sample space or given two sets, select the union, intersection, or complement of two sets.
  • MA.912.DP.4.AP.3 : Given the probability of two events, P(A and B) and P(A), in decimal form, select the conditional probability of the two events {[P(A and B))/(P(A)]}.
  • MA.912.DP.4.AP.6 : Recognize the concept of independence in everyday situations.
  • MA.912.DP.4.AP.7 : Given the probability of two mutually exclusive events in decimal form, use the addition rule for mutually exclusive probabilities: P(A or B)=P(A)+P(B).
  • MA.912.DP.4.AP.8 : Given the probability of two independent events in decimal form, use the multiplication rule for independent probabilities: P(A and B)=P(A)P(B).
  • MA.912.DP.4.AP.2 : Given the probability of events A and B and the product of their probabilities, select whether the events are independent or not independent.
Cluster Resources

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

Lesson Plans
  • What Are the Chances?: Students will develop a program to simulate repeated rolls of a pair of dice in this lesson plan. They will program a realistic interaction between the user and the simulation as well as an analysis tool to identify the theoretical probability and track the observed probability for each outcome.

  • Taxes using Venn Diagrams, Lesson 1:

    Students will review constructing and solving Venn diagrams with two and three data sets. Students will then convert text about the collection of taxes from the local, state, and federal governments into a Venn diagram. This is lesson 1 of a three-part integrated mathematics and civics mini-unit.

  • Taxes using Venn Diagrams, Lesson 2: Students will discuss, recognize, and be challenged to list unions, intersections, and complements related to a Venn diagram created by three data sets. The data is the type of taxes assessed to citizens by the local, state, and federal governments. This is the second lesson in a 3-part integrated mathematics and civic mini-unit.

  • Dropping Out or Staying In: Two-Way Table Analysis: This lesson will require students to calculate relative frequencies and determine if an association exists within a two-way table. The students will analyze the frequencies and write a response justifying the associations and trends found within the table.

  • Breakfast for Champions?: Students will create and interpret two-way frequency tables using joint, marginal, and conditional frequencies in context. They will investigate whether breakfast is for champions.

  • Permutations and Combinations: Students will explore the differences between permutations and combinations. This should follow a lesson on simple probability. This is a great introduction to compound probability and a fun, hands-on activity that allows students to explore the differences between permutations and combinations. This activity leads to students identifying situations involving combinations and permutations in a real-world context.

  • Casino Royale: Students examine games of chance to determine the difference between dependent and independent conditional probability.

  • How to Hit it Big in the Lottery - Probability of Compound Events: Students will explore a wide variety of interesting situations involving probability of compound events. Students will learn about independent and dependent events and their related probabilities.

    Lesson includes:

    • Bell-work that reviews prerequisite knowledge
    • Directions for a great In-Your-Seat Game that serves as an interest builder/introduction
    • Vocabulary
    • Built-in Kagan Engagement ideas
    • An actual lottery activity for real-life application
  • Proposed Budgets: In this Model Eliciting Activity (MEA), students will analyze federal budget data to propose strategic allocations using mathematical skills like expected value calculations and data normalization to justify their recommendations.

    Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

  • Tree Diagrams and Probability: This lesson is designed to develop students' ability to create tree diagrams and figure probabilities of events based on those diagrams. This lesson provides links to discussions and activities related to tree diagrams as well as suggested ways to work them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

  • 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.

  • The Music Is On and Popping! Two-way Tables: This MEA is designed to have teams of 4 students look at data in a two-way table. Teams must discuss which categorical or quantitative factors might be the driving force of a song's popularity. Hopefully, popular songs have some common thread running through them.

    Each team must write down their thought process for creating the most popular playlist of songs for a local radio station. A major constraint for each team is explaining how they will maximize the 11 minutes available with the most popular songs.

    Students will be provided with letters from a local radio station, WMMM - where you can receive your "Daily Mix of Music and Math." WMMM has 10 songs and the researchers have collected data on each. Student teams: it is your responsibility to pick the playlist and write a letter to the station supporting why you made your selection. The winning team gets an opportunity to record a sound bite to introduce their playlist on the radio.

    Now, just when the teams believe they have addressed WMMM's request, a twist is thrown in the midst, and the student teams must return to the drawing board and write a second letter to the station which may or may not affect the team's original playlist.

    Do you have the musical swag to connect the associations?

    Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx.

Perspectives Video: Experts
Problem-Solving Tasks
  • 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.

  • Return to Fred's Fun Factory (with 50 cents): The task is intended to address sample space, independence, probability distributions and permutations/combinations.

  • Lucky Envelopes: Students answer questions about the probabilities of independent and dependent events.

  • Cards and Independence: This problem solving task lets students explore the concept of independence of events.

  • Alex, Mel, and Chelsea Play a Game: This task combines the concept of independent events with computational tools for counting combinations, requiring fluent understanding of probability in a series of independent events.

  • Coffee at Mom's Diner: This task assesses a student's ability to use the addition rule to compute a probability and to interpret a probability in context.

  • 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.

  • Musical Preferences: This problem solving task asks students to make deductions about the kind of music students enjoy by examining data in a two-way table.

  • The Titanic 2: This task lets students explore the concepts of probability as a fraction of outcomes using two-way tables.

  • The Titanic 1: This task asks students to calculate probabilities using information presented in a two-way frequency table.

  • Random Walk III: The task provides a context to calculate discrete probabilities and represent them on a bar graph.

  • Random Walk IV: This problem solving task on probability combinations gives a situation where the numbers are too large to calculate, so abstract reasoning is required in order to compare the different probabilities.

  • The Titanic 3: This problem solving task asks students to determine probabilities and draw conclusions about the survival rates on the Titanic using a table of data.

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.

  • 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.