Lesson 1 | Probability | 10th Grade Mathematics

Objective

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.

Common Core Standards

Core Standards

The core standards covered in this lesson

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").

High School — Statistics and Probability

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").

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.

Foundational Standards

The foundational standards covered in this lesson

7.SP.C.5

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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Describe the sample space as all the outcomes for a particular experiment or situation.
Describe events as subsets of a sample space using characteristics or categories of the outcomes.
Find the probability in the sample space of events that represent “or,” “and,” and “not” situations.
Use the notation $$P(desired \space outcome)$$ to describe probabilities.
Represent the sample space as lists or tree diagrams.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson covers the key concept of representing an experiment or situation with multiple visual representations (lists and tree diagrams). The students should first be given plenty of practice using these models to generate the total number of outcomes of an experiment and then use their own method for representing.
This lesson is an extension of 7.SP.8 standard, which is important because it prepares the students to master the high school statistics and probability standards, specifically under the domain of Conditional Probability and the Rules of Probability. In addition, it may be the case that the students have not studied probability since the seventh grade and need a review.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Ten students on a robotics team fill out an information card describing their gender, grade, whether they are currently taking a science course, whether they are on a sports team, and the number of hours they sleep (on average) per night.

If an information card is chosen at random, which of the following represents the sample space for the event?

$$S=\left\{Student \space ID,gender,grade\space level,science \space class,sports\space team,hours \space of\space sleep\right\}$$
$$S=\left\{ S1,S2,S3,S4,S5,S6,S7,S8,S9,S10 \right\}$$
$$S=\left\{S1,S2,S3,S4,S5,S6,S7,S8,S9,S10,male,female,9,10,11,12,yes,no,6,7,8,9\right\}$$

The teacher in the robotics club wants to know a little more about her team and chooses a card at random. What is the probability that:
1. the selected student is female?
2. the selected student sleeps fewer than 8 hours a night?
3. the selected student is on a sports team or is taking a science class?
4. the selected student is not a junior?

Guiding Questions

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References

Illustrative Mathematics Describing Events

Describing Events, accessed on June 14, 2017, 2:41 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem 2

Annie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated across from each other?

Guiding Questions

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References

Illustrative Mathematics Sitting Across from Each Other

Sitting Across from Each Other, accessed on June 14, 2017, 2:40 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem 3

Sam created a game where he flips a coin and then rolls a die. What is the probability that Sam will flip a head and then roll a 4?

Guiding Questions

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References

NRICH An Introduction to Tree Diagrams

Modified by Fishtank Learning, Inc.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

You flip a coin three times in a row. Your friend tells you that the sample space for your experiment is just “head/tails.” You know he is incorrect.

Represent the sample space of this experiment visually to help convince your friend, and explain your reasoning.
Is there another way to represent this sample space? Why did you choose your representation?

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Illustrative Mathematics Tetrahedral Dice
Illustrative Mathematics Red, Green, or Blue?
Illustrative Mathematics Rolling Twice
EngageNY Mathematics Algebra II > Module 4 > Topic A > Lesson 6 — Opening
Open Middle Probability with Marbles
Open Middle Rolling with the Same Probability
Open Middle Probability of Rolling Two Six Sided Dice

Lesson 2

Lesson Map

Topic A: Conditional Probability and the Rules of Probability

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.

7.SP.C.8 S.CP.A.1

Determine the probability of events with replacement using tree diagrams, addition rules for mutually exclusive events, and multiplication rules for compound events.

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 without replacement using tree diagrams, addition rules for mutually exclusive events, and multiplication rules for compound events.

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 that are not mutually exclusive to formalize the addition rule.

S.CP.A.1 S.CP.A.2 S.CP.B.7

Describe conditional probability and develop the rule $$P(B|A)=\frac{P(A \space \mathrm{and} \space B)}{P(A)}$$.

S.CP.A.3

Determine whether events are independent.

S.CP.A.2 S.CP.A.3 S.CP.A.5

Calculate and analyze relative frequencies in two-way tables to make statements about the data and determine independence.

S.CP.A.4 S.CP.A.5 S.ID.B.5

Make decisions about medical testing based on conditional probabilities.

S.CP.A.3 S.CP.A.4 S.CP.A.5

Describe and apply the counting principle and permutations to contextual and non-contextual situations.

S.CP.B.9

Describe and apply the counting principle and combinations to contextual and non-contextual situations.

S.CP.B.9

Probability

Objective

Common Core Standards

Core Standards

High School — Statistics and Probability

Statistics and Probability

Foundational Standards

Statistics and Probability

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem 3

Guiding Questions

References

Target Task

Additional Practice

Lesson Map

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