## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students understand conditional probability, and, in particular, to help teachers 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.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution-NonCommercial-NoDerivs 3.0
- Media Format:
- Downloadable docs, Text/HTML

## Standards

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret expressions that represent a quantity in terms of its context.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build a function that models a relationship between two quantities

Standard: Write a function that describes a relationship between two quantities.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Factor a quadratic expression to reveal the zeros of the function it defines.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Use the properties of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Statistics and Probability: Conditional Probability and the Rules of ProbabilityCluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model

Standard: (+) Use the rules of probability to compute probabilities of compound events in a uniform probability model. Use permutations and combinations to compute probabilities of compound events and solve problems.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Statistics and Probability: Conditional Probability and the Rules of ProbabilityCluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model

Standard: (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = [P(A)]x[P(B|A)] =[P(B)]x[P(A|B)], and interpret the answer in terms of the model.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Statistics and Probability: Conditional Probability and the Rules of ProbabilityCluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model

Standard: Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Statistics and Probability: Conditional Probability and the Rules of ProbabilityStandard: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Statistics and Probability: Conditional Probability and the Rules of ProbabilityCluster: Understand independence and conditional probability and use them to interpret data

Standard: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Statistics and Probability: Conditional Probability and the Rules of ProbabilityCluster: Understand independence and conditional probability and use them to interpret data

Standard: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Statistics and Probability: Conditional Probability and the Rules of ProbabilityCluster: Understand independence and conditional probability and use them to interpret data

Standard: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Statistics and Probability: Conditional Probability and the Rules of ProbabilityCluster: Understand independence and conditional probability and use them to interpret data

Standard: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Statistics and Probability: Conditional Probability and the Rules of ProbabilityCluster: Understand independence and conditional probability and use them to interpret data

Standard: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as the product of P and a factor not depending on P.*

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret parts of an expression, such as terms, factors, and coefficients.*

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Statistics and Probability: Using Probability to Make DecisionsCluster: Calculate expected values and use them to solve problems

Standard: (+) Calculate expected values and use them to solve problems. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Use the properties of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Factor a quadratic expression to reveal the zeros of the function it defines.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret parts of an expression, such as terms, factors, and coefficients.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as the product of P and a factor not depending on P.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret expressions that represent a quantity in terms of its context.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Write a function that describes a relationship between two quantities.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Statistics and Probability: Conditional Probability and the Rules of Probability

Standard: Use the rules of probability to compute probabilities of compound events in a uniform probability model

Indicator: Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Statistics and Probability: Conditional Probability and the Rules of Probability

Standard: Use the rules of probability to compute probabilities of compound events in a uniform probability model

Indicator: Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Statistics and Probability: Conditional Probability and the Rules of Probability

Standard: Use the rules of probability to compute probabilities of compound events in a uniform probability model

Indicator: (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = [P(A)]x[P(B|A)] =[P(B)]x[P(A|B)], and interpret the answer in terms of the model.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Statistics and Probability: Conditional Probability and the Rules of Probability

Indicator: (+) Use the rules of probability to compute probabilities of compound events in a uniform probability model. Use permutations and combinations to compute probabilities of compound events and solve problems.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Statistics and Probability: Using Probability to Make Decisions

Standard: Calculate expected values and use them to solve problems

Indicator: (+) Calculate expected values and use them to solve problems. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Statistics and Probability: Conditional Probability and the Rules of Probability

Standard: Understand independence and conditional probability and use them to interpret data

Indicator: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or,"ť "and,"ť "not"ť).*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Statistics and Probability: Conditional Probability and the Rules of Probability

Standard: Understand independence and conditional probability and use them to interpret data

Indicator: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Statistics and Probability: Conditional Probability and the Rules of Probability

Standard: Understand independence and conditional probability and use them to interpret data

Indicator: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Statistics and Probability: Conditional Probability and the Rules of Probability

Standard: Understand independence and conditional probability and use them to interpret data

Indicator: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Statistics and Probability: Conditional Probability and the Rules of Probability

Standard: Understand independence and conditional probability and use them to interpret data

Indicator: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize"Óto abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents"Óand the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"ť They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

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# Tags (8)

- Mathematics
- CCSS
- Common Core Math
- Common Core PD
- Conditional Probabilities
- ODE Learning
- Statistics and Probability
- Tree Diagrams

## Comments