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 TITLE : Unit 10: Probability SUGGESTED DURATION : 12 days

#### Unit Overview

Introduction
This unit bundles student expectations that address permutations, combinations, and various types of probability. Concepts are incorporated into both non-contextual and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grade 7, students determined theoretical and experimental probabilities for simple and compound events. Students also described and found the complement of a probability.

During this Unit
Students define and develop formulas for permutations and combinations and apply permutations and combinations to solve contextual problems. Students determine number of possible outcomes of an event, including combinations, permutations, and the Fundamental Counting Principle. Students investigate and define probability in terms of possible outcomes and desired event. Students determine the theoretical and experimental probability based on area models in problem situations. Students identify events as being independent or dependent, and connect independence and dependence to with and without replacement. Students apply the concept of probabilities of independent and dependent events to solve contextual problems. Students are introduced to conditional probability, including notation, and apply conditional probability in contextual problems.

After this Unit
The concepts of permutations, combinations, and probability will be applied in subsequent mathematics courses and real world situations.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): I. Numeric Reasoning A2, B1; II. Algebraic Reasoning D1, D2; III. Geometric and Spatial Reasoning C2; IV. Probabilistic Reasoning A1, B1, B2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to Navigating through Probability in Grades 9 – 12 (2004), from the National Council of Teachers of Mathematics (NCTM), “Probability is an area of the school mathematics curriculum that high school courses often glossed over or sometimes even skip. However, the importance of probabilistic reasoning – particularly, reasoning about the relationship between data and probability – has received increased attention from national groups over the past several decades” (p. 1). Additionally, “The Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) accorded to probability and statistics the same status in the K – 12 mathematics curriculum as algebra, geometry, measurement, and number and operations. Eleven years later, Principals and Standards for School Mathematics (NCTM, 2000), reaffirmed this assessment of the importance of data analysis and probability in the Pre-K – 12 mathematics curriculum” (as cited in NCTM, 2004, p. 1).

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2004). Navigating through probability in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Probability is used to make predictions and inferences about the degree of likelihood of outcomes in everyday life. When is probability applicable in everyday life? Why is it important to be aware of the factors that influence outcomes and how does this understanding affect how one supports or refutes the validity of predictions and inferences made by oneself or others? How can probability be used to reason about uncertain events in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Various methods can be used to determine the total possible outcomes of an event in order to make predictions and critical judgments in problem situations.
• How can …
• combinations
• permutations
… be used to determine the number of ways an event may occur?
• How can it be determined whether to use combinations or permutations?
• Probability is used to make predictions and inferences about outcomes in order to make informed decisions in everyday life.
• What is the purpose of determining the probability of the occurrence of an event?
• How can representations and appropriate language be used to effectively communicate and illustrate probabilistic relationships?
• How is the probability of an event(s) used to make predictions and inferences in problem situations?
• Why are combinations and permutations used in determining probability?
• How is an area probability model applied to solve a real-world problem situation?
• How do experimental and theoretical probability compare?
• How does the number of trials in an experiment affect the validity of the experimental probability?
• What conditions might necessitate the use of …
• theoretical probability?
• experimental probability?
• What tools and processes can be used to determine the probability of the occurrence of …
• events?
• compound events, including dependent and independent events?
• events involving conditional probability?
• How are compound events distinguished as independent or dependent?
• Probability
• Counting Principles
• Permutations
• Combinations
• Formulas
• Probability
• Area probability model
• Conditional
• Formulas
• Probability of Events
• Dependence
• Independence
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may confuse when to use permutations and when to use combinations to determine outcomes.
• Some students may not realize that an event is a subset of the total possible outcomes.
• Some students may confuse independent events and dependent events.
• Some students may think that if objects are not replaced in the set it does not impact the probability instead of realizing that the total outcomes for the next probability will be decreased.

#### Unit Vocabulary

• Combinations – number of different ways a set of objects can be selected without regard to a specific order
• Complement of an event – the probability of the non-occurrence of a desired outcome; the sum of the probability of the event and the probability of the non-occurrence of the event are equal to one
• Compound events – a set of outcomes from a combination of actions or activities where the outcomes can be subdivided (e.g., flipping a coin and rolling a number cube, drawing tiles out of a bag and spinning a spinner, etc.)
• Conditional probability – for two events, A and B, the probability of B given that A has already occurred; written as P(B|A); read as “the probability of B, given A”
• Dependent events – the outcome from one action or activity may affect the probability of the outcome(s) of any subsequent action(s) or activity(s); usually involves compound events
• Experimental probability – the likelihood of an event occurring from the outcomes of an experiment
• Independent events – the outcome from one action or activity does not affect the probability of the outcome(s) of any subsequent action(s) or activity(s); usually involves compound events
• Independent probability – if P(B|A) = P(B), then A and B are said to be independent
• Permutations – number of different ways a set of objects can be selected with regard to a specific order
• Probability – a ratio between the number of desired outcomes to the total possible outcomes, 0 ≤ p ≤ 1
• Sample space – a set of all possible outcomes of one or more events
• Theoretical probability – the likelihood of an event occurring predicted by using formulas and mathematical calculations without conducting an experiment

Related Vocabulary:

 Equally likely Event Experiment Factorial Favorable outcomes Outcome Tree diagram
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Geometry Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
G.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
G.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
G.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
G.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
G.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
G.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
G.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
•  VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
G.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII. C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
G.13 Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events. The student is expected to:
G.13A Develop strategies to use permutations and combinations to solve contextual problems.

Develop

STRATEGIES TO USE PERMUTATIONS AND COMBINATIONS

Including, but not limited to:

• Permutations – number of different ways a set of objects can be selected with regard to a specific order
• Formula for permutations: , where n is the total number of objects in the set and r is the number to be chosen.
• Combinations – number of different ways a set of objects can be selected without regard to a specific order
• Formula for combinations: , where n is the total number of objects in the set and r is the number to be chosen.
• Strategies to determine permutations and combinations
• Diagram
• Lists
• Formulas
• Graphing calculator technology

To Solve

CONTEXTUAL PROBLEMS

Including, but not limited to:

• Application of permutations to determine the number of ways an event can occur in contextual problem situations
• Application of combinations to determine the number of ways an event can occur in contextual problem situations

Note(s):

• Grade 7 introduced simple and compound events and applications of probability.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• IV.A. Probabilistic Reasoning – counting principles
• IV.A.1. Determine the nature and the number of elements in a finite sample space.
G.13B Determine probabilities based on area to solve contextual problems.

Determine

PROBABILITIES BASED ON AREA TO SOLVE CONTEXTUAL PROBLEMS

Including, but not limited to:

• Probability – a ratio between the number of desired outcomes to the total possible outcomes, 0 ≤ p ≤ 1
• Sample space – a set of all possible outcomes of one or more events
• Experimental probability – the likelihood of an event occurring from the outcomes of an experiment
• Theoretical probability – the likelihood of an event occurring predicted by using formulas and mathematical calculations without conducting an experiment
• Complement of an event – the probability of the non-occurrence of a desired outcome; the sum of the probability of the event and the probability of the non-occurrence of the event are equal to one
• Representation of probability with an area model
• Representation of real-world problem situations with area models
• Representation of data sets with an area model (circle graph)
• Determination of probability from an area model
• Hands-on and technology to model and simulate events

Note(s):

• Grade 7 introduced simple and compound events and applications of probability.
• Geometry extends the concept of probabilities based on area.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.2. Make connections between geometry, statistics, and probability.
• IV.B. Probabilistic Reasoning – Computation and interpretation of probabilities
• IV.B.1. Compute and interpret the probability of an event and its complement.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
G.13C Identify whether two events are independent and compute the probability of the two events occurring together with or without replacement.

Identify

WHETHER TWO EVENTS ARE INDEPENDENT

Including, but not limited to:

• Compound events – a set of outcomes from a combination of actions or activities where the outcomes can be subdivided (e.g., flipping a coin and rolling a number cube, drawing tiles out of a bag and spinning a spinner, etc.)
• Independent events – the outcome from one action or activity does not affect the probability of the outcome(s) of any subsequent action(s) or activity(s); usually involves compound events(s)
• Dependent events – the outcome from one action or activity may affect the probability of the outcome(s) of any subsequent action(s) or activity(s); usually involves compound events

Compute

THE PROBABILITY OF THE TWO EVENTS OCCURRING TOGETHER WITH OR WITHOUT REPLACEMENT

Including, but not limited to:

• Probability – a ratio between the number of desired outcomes to the total possible outcomes, 0 ≤ p ≤ 1
• Compound independent events with replacement
• Compound dependent events without replacement

Note(s):

• Grade 7 introduced simple and compound events and applications of probability.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• IV.B. Probabilistic Reasoning – Computation and interpretation of probabilities
• IV.B.1. Compute and interpret the probability of an event and its complement.
• IV.B.2. Compute and interpret the probability of conditional and compound events.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
G.13D Apply conditional probability in contextual problems.

Apply

CONDITIONAL PROBABILITY IN CONTEXTUAL PROBLEMS

Including, but not limited to:

• Conditional probability – for two events, A and B, the probability of B given that A has already occurred; written as P(B|A); read as “the probability of B, given A”
• P(B|A) = • Application of conditional probability in contextual problems

Note(s):

• Grade 7 introduced simple and compound events and applications of probability.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• IV.B. Probabilistic Reasoning – Computation and interpretation of probabilities
• IV.B.1. Compute and interpret the probability of an event and its complement.
• IV.B.2. Compute and interpret the probability of conditional and compound events.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
G.13E Apply independence in contextual problems.

Apply

INDEPENDENCE IN CONTEXTUAL PROBLEMS

Including, but not limited to:

• Application of independence in contextual problems.
• Independent events in terms of conditional probability
• Conditional probability – for two events, A and B, the probability of B given that A has already occurred; written as P(B|A); read as “the probability of B, given A”
• Independent probability – if P(B|A) = P(B), then A and B are said to be independent
• Distinguishing between events that are independent and dependent using contextual problems involving conditional probability
• Computing compound probabilities based on conditional events, using formulas such as P(A and B) = P(A) • P(B|A).

Note(s): 