Lesson 9 | Probability | 10th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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

High School — Statistics and Probability

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

Criteria for Success

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

Describe that a permutation is a sample space where the order in which the events occur matters.
Calculate the total number of permutations when repetition is allowed by calculating $$ n^x$$, where $$x$$ is the number of positions you have available and $$n$$ is the number of choices you have for each of those positions. (This is a form of the counting principle.)
Calculate the total number of permutations when repetition is not allowed by calculating $$n!$$, where $$n$$ is the number of choices you have for each position. $$ n! $$ Is represented algebraically as $$(n)(n-1)(n-2)(n-3)…$$ until the last binomial is equal to $$1$$.
Calculate the permutations when choosing $$k$$ from a set of $$n$$ items—that is, given $$k$$ different positions available and $$n$$ set of items where $$ k<n$$, use the formula $$_nP_k=\frac{n!}{(n-k)!}$$.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson covers S-CP.9 (+) standard, which is a plus standard and therefore an optional lesson in this unit.
It may seem counterintuitive to study permutations before combinations, but the formulas for calculation build from permutations (where order is important) to combinations (where order is not important).
Anchor Problem #2 and Anchor Problem #3 build upon one another but are separated into two different problems because they serve different goals.
EXTENSION: Use dy/dan, a blog by Dan Meyer, “My Lesson Plan: The Door Lock” to do a larger task than Anchor Problem #2 and Anchor Problem #3, where there is only a five-number combination lock.

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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

Using the digits 1, 2, and 3, write an expression to answer the following questions

How many different three-digit numbers can you make when you can use each digit more than once?
How many different three-digit numbers can you make when you can only use each digit once?

Guiding Questions

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Problem 2

Your phone requires you to use five numbers for the code, but you only can use numbers 1-5 for the code.

The first combination you come up with is:

You then realize how "insecure" this number might be since anyone could guess it. Figuring this out you wonder, "How many possibilities are there for combinations if I only use each number once?"

Guiding Questions

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Problem 3

You change the phone settings so that you only need a three-digit code to open the lock.

This means you have 5 numbers to choose from and 3 positions to put those numbers in, which is represented by $$_5P_3$$. How many different combinations can you get if you have three positions and a choice of five numbers with no repetition?

Guiding Questions

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Target Task

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

Problem 1

The combination for a lock consists of three numbers.

If the numbers can be repeated, how many different combinations are there? Explain your answer.
If the numbers cannot be repeated, how many different combinations are there? Explain your answer.

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 2 — Exit Ticket, Question #1

Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 2 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.

Problem 2

Jacqui is putting together sets of greeting cards for a school fundraiser. There are four different card options, two different colored envelopes, and four different sticker designs. A greeting card set consists of one type of card, one color for the envelopes, and one sticker design. How many different ways can Jacqui arrange the greeting card sets? Explain how you determined your answer.

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 2 — Exit Ticket, Question #2

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 that ask for a probability of choosing the right random permutation, applying a “time per permutation,” and identifying the amount of time. (Using a movie clip where a device is cracking an alarm system code (how fast is the machine working) would be ideal.)

Math Misery? "Medical", claimed maliced decimal
EngageNY Mathematics Precalculus and Advanced Topics > Module 5 > Topic A > Lesson 2
JMAP Regents Exam Questions S.CP.B.9: Permutations 2

Lesson 8

Lesson 10

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

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Target Task

Problem 1

References

Problem 2

References

Additional Practice

Lesson Map

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