Probability
Students formalize their understanding of compound probability, develop an understanding of conditional probability, and understand and calculate permutations and combinations.
Unit Summary
In Unit 8, Probability, students extend their understanding of probability from seventh grade to a more formal approach to probability by applying formulas and definitions.
Students begin the unit using visual representations of lists, tree diagrams, and Venn diagrams to find the probability of events that intersect or represent the union or complement of outcomes. Students also formalize their understanding of compound probability. Then, students develop a conceptual and procedural understanding of conditional probability and how this can be used to determine whether variables are independent. In application, students use this knowledge to solve a set of applications on medical testing. Finally, as an optional part of the unit (covering plus (+) Common Core State Standards), students understand and calculate permutations and combinations. The students develop an understanding that whether order matters or not can affect the sample space of the total combinations or the number of chosen items within a set of items.
In Algebra 2, students will continue their study of probability by studying statistical inference and making decisions using probability.
Pacing: 12 instructional days (10 lessons, 1 flex day, 1 assessment day)
Assessment
This assessment accompanies Unit 8 and should be given on the suggested assessment day or after completing the unit.
Unit Prep
Intellectual Prep
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Internalization of Standards via the Unit Assessment
- Take unit assessment. Annotate for:
- Standards that each question aligns to
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) that assessment points to
Internalization of Trajectory of Unit
- Read and annotate the Unit Summary.
- Notice the progression of concepts through the unit using the Lesson Map.
- Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions
Essential Understandings
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- Probability helps us to analyze the chance that an event occurs, and it provides a framework with which to make decisions about future events based on known information.
- Determining independence or dependence of variables through analysis of the probabilities is essential to drawing conclusions about a random experiment or observational study.
- Understanding the sample space of an event, that is, how many total combinations or permutations exist, ensures that all the intricacies of a situation are accounted for.
Vocabulary
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| sample space | $$P$$ (desired outcome) |
| complement | tree diagram |
| with replacement | without replacement |
| mutually exclusive (disjoint) | addition rule |
| multiplication rule | equally likely (fair) |
| compound probability | venn diagram |
| conditional probability | dependent event |
| independent event | relative frequency |
| two-way table | medical testing |
| false positive/false negative | true positive/true negative |
| permutation | combination |
| fundamental counting principle |
Materials
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- Dice
- Cards
- Spinners
- Coins
Lesson Map
Topic A: Conditional Probability and the Rules of Probability
1
7.SP.C.8
S.CP.A.1
Describe the sample space of an experiment or situation. Use probability notation to identify the “and,” “or,” and complement outcomes from a given sample space.
2
7.SP.C.7
S.CP.A.2
S.CP.A.4
S.CP.B.6
S.CP.B.7
Determine the probability of events with replacement using tree diagrams, addition rules for mutually exclusive events, and multiplication rules for compound events.
3
7.SP.C.7
S.CP.A.2
S.CP.B.6
S.CP.B.7
S.CP.B.8
Determine the probability of events without replacement using tree diagrams, addition rules for mutually exclusive events, and multiplication rules for compound events.
4
S.CP.A.1
S.CP.A.2
S.CP.B.7
Determine the probability of events that are not mutually exclusive to formalize the addition rule.
5
S.CP.A.3
Describe conditional probability and develop the rule $$P(B|A)=\frac{P(A \space \mathrm{and} \space B)}{P(A)}$$.
6
S.CP.A.2
S.CP.A.3
S.CP.A.5
Determine whether events are independent.
7
S.ID.B.5
S.CP.A.4
S.CP.A.5
Calculate and analyze relative frequencies in two-way tables to make statements about the data and determine independence.
8
S.CP.A.3
S.CP.A.4
S.CP.A.5
Make decisions about medical testing based on conditional probabilities.
9
S.CP.B.9
Describe and apply the counting principle and permutations to contextual and non-contextual situations.
10
S.CP.B.9
Describe and apply the counting principle and combinations to contextual and non-contextual situations.
Common Core Standards
Key: Major Cluster Supporting Cluster Additional Cluster
Core Standards
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High School — Statistics and Probability
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S.CP.A.1 — 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").
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S.CP.A.2 — 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.
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S.CP.A.3 — 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.
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S.CP.A.4 — 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.
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S.CP.A.5 — 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.
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S.CP.B.6 — 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.
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S.CP.B.7 — 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.
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S.CP.B.8 — Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
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S.CP.B.9 — Use permutations and combinations to compute probabilities of compound events and solve problems.
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S.ID.B.5 — Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
Statistics and Probability
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7.SP.C.7 — Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
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7.SP.C.8 — Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
Foundational Standards
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Statistics and Probability
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7.SP.C.5
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7.SP.C.6
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8.SP.A.4
Future Standards
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High School — Statistics and Probability
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S.IC.A.2
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S.IC.B.5
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S.IC.B.6
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S.MD.A.1
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S.MD.A.2
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S.MD.A.3
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S.MD.A.4
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S.MD.B.7
Using Probability to Make Decisions
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S.MD.B.5
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S.MD.B.6
Standards for Mathematical Practice
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CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
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CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
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CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
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CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
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CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
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CCSS.MATH.PRACTICE.MP6 — Attend to precision.
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CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
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CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.