Math / 7th Grade / Unit 8: Probability

Probability

Students have their first encounter with probability, as they develop their understanding of probability through calculating theoretical probabilities and designing and running their own simulations.

Math

Unit 8

7th Grade

Unit Summary

In Unit 8, seventh-grade students finish the year with their first encounter with probability. They develop their understanding of probability through analyzing experiments, calculating theoretical probabilities, and designing and running their own simulations to model real-world situations (MP.4). Students encounter and use a variety of tools including spinners, dice, cards, coins, etc., and organizational tools such as organized lists, tables, and tree diagrams when they study compound probability (MP.5).

Students draw on and re-engage with concepts from ratios and proportions in order to fully understand probability as a ratio of desired outcomes to total outcomes. They also use proportional relationships to estimate long-run frequencies based on probabilities of experiments.

In high school, students will further explore probability, distinguishing between independent events and conditional events and developing rules to calculate probabilities of these compound events.

Pacing: 11 instructional days (9 lessons, 1 flex day, 1 assessment day)

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Assessment

The following assessments accompany Unit 8.

Pre-Unit

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

Mid-Unit

Have students complete the Mid-Unit Assessment.

Post-Unit

Use the resources below to assess student mastery of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Internalization of Standards via the Post-Unit Assessment

Take the Post-Unit Assessment. Annotate for:
- Standards that each question aligns to
- 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 Post-Unit Assessment questions
Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

Unit-Specific Intellectual Prep

Read the Progressions for the Common Core State Standards in Mathematics, 6-8 Statistics and Probability for standards relevant to this unit.
Read the following table that includes examples of models used in this unit.

Model	Example
Organized list	The sample space for a spinner with 4 equal sections labeled yellow, blue, green, and red, and a fair coin with sides heads and tails
Table	The sample space for a spinner with 4 equal sections labeled yellow, blue, green, and red, and a fair coin with sides heads and tails
Tree diagram	The sample space for a spinner with 4 equal sections labeled yellow, blue, green, and red, and a fair coin with sides heads and tails

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

The probability of an event or combination of events occurring is determined by the number of desired or favorable outcomes divided by the total number of outcomes possible. This value ranges from 0, where the event is impossible, to 1, where the event is certain to occur, with various levels of likelihood in between.
Experimental or theoretical probabilities can be used to estimate or predict long-run frequencies.
Real-world situations can be simulated using various probability models in order to test hypotheses or make predictions based on data.

Materials

The materials, representations, and tools teachers and students will need for this unit

Dice (2 per student)
Counters (11 per small group)
Brown paper bag (2 per pair of students)
Cubes (1 set per small group) (15 with 5 red and 10 of different colors)
Plastic cups (non-see-through) (3 per small group)
Plastic balls (like ping-pong) (1 per small group)
Coins (1 per student)
Optional: Standard deck of playing cards (1 per small group)
Optional: Spinners
The Horse Race Board (1 per small group)
Race Results Sheet (1 per small group)

Vocabulary

Terms and notation that students learn or use in the unit

compound event

experimental probability

likelihood

outcome

probability

sample space

simulation

simple event

theoretical probability

tree diagram

To see all the vocabulary for Unit 8, view our 7th Grade Vocabulary Glossary.

Lesson Map

Topic A: Probability Models of Simple Events

Understand the probability of an event happening is a number between 0 and 1, ranging from impossible to certain.

7.SP.C.5

Define probability and sample space. Estimate probabilities from experimental data.

7.SP.C.6 7.SP.C.7

Determine the probability of events.

7.SP.C.7.A 7.SP.C.7.B

Use probability to predict long-run frequencies.

7.SP.C.6

Design and conduct simulations to model real-world situations.

7.SP.C.7

Topic B: Probability Models of Compound Events

Conduct simulations with multiple events to determine probabilities.

7.SP.C.8 7.SP.C.8.C

List the sample space for compound events using organized lists, tables, or tree diagrams.

7.SP.C.8.B

Determine the probability of compound events.

7.SP.C.8

Design and conduct simulations to model real-world situations for compound events.

7.SP.C.8.C

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Statistics and Probability

7.SP.C.5 — Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Statistics and Probability

7.SP.C.5 — Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7.SP.C.6 — Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

Statistics and Probability

7.SP.C.6 — Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
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.

Statistics and Probability

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.
7.SP.C.7.A — Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

Statistics and Probability

7.SP.C.7.A — Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
7.SP.C.7.B — Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

Statistics and Probability

7.SP.C.7.B — Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7.SP.C.8 — Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

Statistics and Probability

7.SP.C.8 — Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
7.SP.C.8.A — Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

Statistics and Probability

7.SP.C.8.A — Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
7.SP.C.8.B — Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.

Statistics and Probability

7.SP.C.8.B — Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
7.SP.C.8.C — Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

Statistics and Probability

7.SP.C.8.C — Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

Number and Operations—Fractions

5.NF.A.2

Number and Operations—Fractions

5.NF.A.2 — Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Ratios and Proportional Relationships

7.RP.A.2

Ratios and Proportional Relationships

7.RP.A.2 — Recognize and represent proportional relationships between quantities.
7.RP.A.3

Ratios and Proportional Relationships

7.RP.A.3 — Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Statistics and Probability

7.SP.A.1

Statistics and Probability

7.SP.A.1 — Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

Future Standards

Standards in future grades or units that connect to the content in this unit

Conditional Probability and the Rules of Probability

HSS-CP.A.1

Conditional Probability and the Rules of Probability

HSS-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").
HSS-CP.A.2

Conditional Probability and the Rules of Probability

HSS-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.
HSS-CP.A.3

Conditional Probability and the Rules of Probability

HSS-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.
HSS-CP.A.4

Conditional Probability and the Rules of Probability

HSS-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.
HSS-CP.A.5

Conditional Probability and the Rules of Probability

HSS-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.
HSS-CP.B.6

Conditional Probability and the Rules of Probability

HSS-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.
HSS-CP.B.7

Conditional Probability and the Rules of Probability

HSS-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.
HSS-CP.B.8

Conditional Probability and the Rules of Probability

HSS-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.
HSS-CP.B.9

Conditional Probability and the Rules of Probability

HSS-CP.B.9 — Use permutations and combinations to compute probabilities of compound events and solve problems.

Statistics and Probability

8.SP.A.4

Statistics and Probability

8.SP.A.4 — Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

Using Probability to Make Decisions

HSS-MD.A.3

Using Probability to Make Decisions

HSS-MD.A.3 — Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
HSS-MD.A.4

Using Probability to Make Decisions

HSS-MD.A.4 — Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Unit 7

Statistics