Lesson 1 | Probability and Statistical Inference | 11th Grade Mathematics

Objective

Determine probabilities of mutually exclusive events.

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").

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.
7.SP.C.6

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

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.8

Statistics and Probability

7.SP.C.8 — Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

Criteria for Success

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

Describe a probability as a number between $$0$$ and $$1$$ used to quantify uncertain outcomes and that the sum of all possible outcomes is $$1$$.
Use notation of $${P(\mathrm{desired \space outcome})}$$ to describe probabilities.
Describe that when $$P(A\space \mathrm{and} \space B)=0$$, outcomes $$A$$ and $$B$$ are mutually exclusive and cannot both occur.
Use a tree diagram to describe the sample space of an experiment where events are mutually exclusive and identify $$P(A), \space P(A \space \mathrm{or} \space B), \space O(\mathrm{not}\space A), \space P(A,B)$$.
Calculate the probability of mutually exclusive events $$A$$ and $$B$$ as $$P(A\space\mathrm{or}\space B)=P(A)+P(B)$$.
Calculate the probability of mutually exclusive events $$A$$ and $$B$$ as $$P(A,B)=P(A)\times P(B)$$.
Describe the difference between dependent outcomes and independent outcomes for the case of replacement.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Here is a great spinner simulation by NCTM Illuminations. This can be run in the background while you teach or you can “skip to the end” for short-run and long-run trials so kids can see the difference. We will represent long-run probabilities further on in the unit when we discuss normal distributions.
The question for Anchor Problem #2 is just a stepping-off point. The guiding questions for Anchor Problem #1 are key to remembering and synthesizing basic probability terms and concepts.
Here are two sources to understand mutually exclusive events:
- Mutually Exclusive Events by Math Is Fun.
- Independent and Mutually Exclusive Events by The Math Forum, Ask Dr. Math.
This lesson deals with independent events (Anchor Problem #2) and dependent events (Anchor Problem #3). If this lesson feels too long for one class period, split the lesson so that Anchor Problem #2 is taught in one period and Anchor problem #3 is taught in the other.

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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

Each scenario on the left corresponds with a calculation on the right. Connect the scenarios and calculations and identify which features are most useful in connecting the representations:

Rolling once the number is an odd number or a number less than 3.	$${P(A\space\mathrm{and}\space B) = \left({3\over6}\right)\left({1\over6}\right)}$$
Rolling once and the number is an odd number or 4.	$${P(A\space\mathrm{or}\space B) = {3\over6}+{2\over6}-{1\over6}}$$
Rolling twice. The first roll is an odd number and the second roll is a number less than 3.	$${P(A\space\mathrm{and}\space B) = \left({3\over6}\right)\left({2\over6}\right)}$$
Rolling twice. The first roll is an odd number and the second roll is a 4.	$${P(A\space\mathrm{or}\space B) = {3\over6}+{1\over6}}$$

Guiding Questions

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References

New Visions for Public Schools Probability Rules of Addition and Multiplication

Probability Rules of Addition and Multiplication is made available by New Visions for Public Schools under the CC BY-NC-SA 4.0 license. © 2017 New Visions for Public Schools. Accessed https://curriculum.newvisions.org/math/resources/resource/algebra-ii-unit-5-big-idea-1-probability-rules-of-addition-and-multiplication/.

Modified by Fishtank Learning, Inc.

Problem 2

Below is a spinner with four quadrants, each labeled 1–4. Each outcome is equally likely.

Sam spins the spinner twice and doesn’t land on 4. What is the probability of this occurring?

Guiding Questions

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Problem 3

Allison has put together a simple game. She put 6 cubes in a paper bag—3 yellow and 3 blue. Allison has determined that the rules are as follows:

Pull a cube from the bag and put it on the table.
Pull a second cube from the bag.
If the cubes are different colors, then Player A wins.
If the cubes are the same color, then Player B wins.

Is this game fair? How do you know?

Guiding Questions

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References

MARS Formative Assessment Lessons for High School Representing Conditional Probabilities 1 — "Lucky Dip"

Representing Conditional Probabilities 1 from the Classroom Challenges by the MARS Shell Center team at the University of Nottingham is made available by the Mathematics Assessment Project under the CC BY-NC-ND 3.0 license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed June 15, 2017, 1:49 p.m..

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

Use the scenario below to answer the questions that follow.

Game Tools:

Spinner 1 (three equal sectors with the number 1 in one sector, the number 2 in the second sector, and the number 3 in the third sector)
Card bag (Blue-A, Blue-B, Blue-C, Blue-D, Red-E, Red-F)

Directions (chance experiment): Spin Spinner 1, and randomly select a card from the card bag (four blue cards and two red cards). Record the number from your spin and the color of the card selected.

For the chance experiment described in the scenario, why is the probability of the event "spinning an odd number and randomly selecting a blue card" not the same as the probability of the event "spinning an even number and randomly selecting a blue card"? Which event would have the greater probability of occuring, and why?
Why is the probability of the event "spinning an odd number from Spinner 1 and randomly selecting a blue card" not equal to the probability of "spinning an odd number from Spinner 1 or randomly selecting a blue card"?

References

EngageNY Mathematics Algebra II > Module 4 > Topic A > Lesson 1 — Exit Ticket, Questions #1 and #2

Algebra II > Module 4 > Topic A > Lesson 1 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

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.

Include problems varying representations – spinners, coins, dice, and more random situations.
Include problems emphasizing the difference between $${P(A\space \mathrm{or}\space B)=P(A)+P(B)}$$ and $${P(A,B) = P(A) \times P(B)}$$.

Illustrative Mathematics But Mango Is My Favorite...
EngageNY Mathematics Algebra II > Module 4 > Topic A > Lesson 1 — Problem Set (Focus on only mutually exclusive events - no Venn diagrams)
Illustrative Mathematics 7.SP.C Problems — (Use practice to reinforce background knowledge even though they are not directly tied to the standards for this unit; emphasize the visual model and connection to the addition and multiplication rules)

Lesson 2

Lesson Map

Topic A: Probability

Determine probabilities of mutually exclusive events.

S.CP.A.1

Determine probabilities of events that are not mutually exclusive.

S.CP.A.1 S.CP.B.6 S.CP.B.7

Calculate conditional probabilities.

S.CP.A.3

Determine when events are independent and describe independent events using everday language.

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

Calculate relative frequencies in two-way tables to analyze data and determine independence.

S.CP.A.4

Use conditional probability to make decisions about medical testing.

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

Topic B: The Normal Distribution

Describe the center, shape, and spread of distributions by reasoning visually about the mean, standard deviation, and shape of a histogram.

S.IC.A.1

Derive and calculate population percentages based on a normal distribution of data.

S.IC.A.2 S.IC.B.4

Use $${z-}$$scores to identify population percentiles.

S.IC.B.4

Topic C: Statistical Inferences and Conclusions

Describe and compare statistical study methods.

S.IC.B.3 S.IC.B.6

Use multiple random samples to estimate a population mean or proportion and verify the validity of the sampling method by analyzing the means and standard errors of samples.

S.IC.B.4

Calculate and describe the margin of error in context and use larger sample sizes to minimize the margin of error.

S.IC.B.4

Compare two treatments in experimental data and determine if the difference between the two treatments is significant.

S.IC.B.5

Probability and Statistical Inference

Objective

Common Core Standards

Core Standards

High School — Statistics and Probability

Foundational Standards

Statistics and Probability

Statistics and Probability

Statistics and Probability

Statistics and Probability

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Target Task

References

Additional Practice

Lesson Map

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