correctionkey=nl-b;ca-b correctionkey=nl-c;ca-c 19 … · explore working with sets ... number on a...

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Explore Working with SetsA set is a collection of distinct objects. Each object in a set is called an element of the set. A set is often denoted by writing the elements in braces.

The set with no elements is the empty set, denoted by ⌀ or { }.

The set of all elements under consideration is the universal set, denoted by U.

Identifying the number of elements in a set is important for calculating probabilities.

A Use set notation to identify each set described in the table and identify the number of elements in each set.

Set Set NotationNumber of Elements in the Set

Set A is the set of prime numbers less than 10.

A = ⎧

⎨ ⎩ 2, 3, , 7

⎫

⎬ ⎭ n (A) = 4

Set B is the set of even natural numbers less than 10.

B = ⎧

⎨ ⎩ , , ,

⎫ ⎬ ⎭ n (B) =

Set C is the set of natural numbers less than 10 that are multiples of 4.

= ⎧

⎨ ⎩ 4,

⎫ ⎬ ⎭ n (C) =

The universal set is all natural numbers less than 10.

U = ( ) = 9

5

2

C 8

n U

2

4

⎧ ⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 8, 9

⎫ ⎬ ⎭

4 6 8

Module 19 949 Lesson 1

19.1 Probability and Set TheoryEssential Question: How are sets and their relationships used to calculate probabilities?

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A2_MNLESE385900_U8M19L1.indd 949 8/26/14 5:47 PM

Common Core Math StandardsThe student is expected to:

S-CP.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”).

Mathematical Practices

MP.6 Precision

Language ObjectiveExplain to a partner how to find the probability of rolling a certain number on a number cube and how to find its complement.

HARDCOVER PAGES 689698

Turn to these pages to find this lesson in the hardcover student edition.

Probability and Set Theory

ENGAGE Essential Question: How are sets and their relationships used to calculate probabilities?To calculate the probability of an event, you need to

know the number of items in the set of outcomes

for that event, as well as the number of items in the

set of all possible outcomes. The theoretical

probability of the event is the ratio of the two

numbers.

PREVIEW: LESSONPERFORMANCE TASKView the Engage section online. Discuss the photograph. Ask students to describe math problems that could be illustrated by the two dogs. Then preview the Lesson Performance Task.

949

HARDCOVER PAGES 689698

Turn to these pages to find this lesson in the hardcover student edition.

S-CP.1 For the full text of this standard, see the table starting on page CA2 of Volume 1.

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Explore Working with Sets

A set is a collection of distinct objects. Each object in a set is called an element of the set. A set

is often denoted by writing the elements in braces.

The set with no elements is the empty set, denoted by ⌀ or { }.

The set of all elements under consideration is the universal set, denoted by U.

Identifying the number of elements in a set is important for calculating probabilities.

Use set notation to identify each set described in the table and identify the number of

elements in each set.

SetSet Notation

Number of

Elements in

the Set

Set A is the set of prime

numbers less than 10.

A = ⎧

⎨ ⎩ 2, 3, , 7

⎫

⎬ ⎭ n (A) = 4

Set B is the set of even natural

numbers less than 10.

B = ⎧

⎨ ⎩ , , ,

⎫

⎬ ⎭ n (B) =

Set C is the set of natural

numbers less than 10 that are

multiples of 4.

= ⎧

⎨ ⎩ 4,

⎫

⎬ ⎭

n (C) =

The universal set is all natural

numbers less than 10.U =

( ) = 9

5

2

C8

n U

2

4

⎧ ⎨ ⎩ 1

, 2, 3, 4, 5, 6, 7, 8, 9 ⎫ ⎬ ⎭

4 6 8

Module 19

949

Lesson 1

19 . 1 Probability and Set Theory

Essential Question: How are sets and their relationships used to calculate probabilities?

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A2_MNLESE385900_U8M19L1 949

8/26/14 8:15 PM

949 Lesson 19 . 1

L E S S O N 19 . 1

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CB

U

A B

U

4,8 2,6

1,3,5,7,9

A B

U

4,6,83,5,7

1,9

2

3,5,7 4,6,8

1,9

2

A

U

2,3,5,7

1,4,6,8,9

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The following table identifies terms used to describe relationships among sets. Use sets A, B, C, and U from the previous table. You will supply the missing Venn diagrams in the Example column, including the referenced elements of the sets, as you complete steps B–I following.

Term Notation Venn Diagram Example

Set C is a subset of set B if every element of C is also an element of B.

C ⊂ B

The intersection of sets A and B is the set of all elements that are in both A and B.

A ⋂ B

A ⋂ B is the double-shaded region.

The union of sets A and B is the set of all elements that are in A or B.

A ⋃ B

A ⋃ B is the entire shaded region.

The complement of set A is the set of all elements in the universal set U that are not in A.

A C or ∼A

AC is the shaded region.

B Since C is a subset of B, every element of set C, which consists of the numbers and , is located not only in oval C, but also within oval B. Set B includes the elements of C as well as the additional elements and , which are located in oval B outside of oval C. The universal set includes the elements of sets B and C as well as the additional elements , , , , and , which are located in region U outside of ovals B and C.

C In the first row of the table, draw the corresponding Venn diagram that includes the elements of B, C, and U.

D To determine the intersection of A and B, first define the elements of set A and set B separately, then identify all the elements found in both sets A and B.

A = ⎧

⎨ ⎩ , , ,

⎫

⎬ ⎭

B = ⎧

⎨ ⎩ , , ,

⎫

⎬ ⎭

A ⋂ B = ⎧

⎨ ⎩

⎫

⎬ ⎭

4

1 35 7 9

2

8

6

See first row of table.

2

2

2

3

4

5

6

7

8




A2_MNLESE385900_U8M19L1 950 8/26/14 8:55 PM

EXPLORE Working with Sets

INTEGRATE TECHNOLOGYStudents have the option of doing the Explore activity either in the book or online.

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Discuss how the Venn diagrams provide pictures of set relationships to help students understand the terminology. Encourage students to practice drawing Venn diagrams to use when investigating set theory. For example, discuss how a Venn diagram can make it easier to identify the complement of an intersection.

AVOID COMMON ERRORSStudents may assume that a set that contains only 0 is the same as the empty set. Contrast the empty set, { }, with the set that contains only 0, {0}.

QUESTIONING STRATEGIESWhat word corresponds to the intersection of two sets? Is it union? Explain. And means the

elements are in both sets, which corresponds to the

intersection. Or means the elements can be in either

set, which corresponds to the union.

How is an intersection different from a subset? The intersection consists of the

elements two sets have in common, while all of the

elements of a subset lie within the set of which it is a

subset.

How do you know when sets overlap? Sets

will overlap when they have some elements

in common.

PROFESSIONAL DEVELOPMENT

Math BackgroundA German mathematician, Georg Cantor (1845–1918), is considered to be the father of set theory. Cantor discovered that the rational numbers are countable but the real numbers are uncountable.

Two French mathematicians, Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665), are considered to be the founders of probability theory. The roots of probability theory lie in the letters they exchanged analyzing games of chance.

Probability and Set Theory 950


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E In the second row of the table, draw the Venn diagram for A ⋂ B that includes the elements of A, B, and U and the double-shaded intersection region.

F To determine the union of sets A and B, identify all the elements found in either set A or set B by combining all the elements of the two sets into the union set.

A ⋃ B = ⎧

⎨ ⎩ , , , , , ,

⎫

⎬ ⎭

G In the third row of the table, draw the Venn diagram for A ⋃ B that includes the elements of A, B, and U and the shaded union region.

H To determine the complement of set A, first identify the elements of set A and universal set U separately, then identify all the elements in the universal set that are not in set A.

A = ⎧

⎨ ⎩ , , ,

⎫

⎬ ⎭

U = ⎧

⎨ ⎩ , , , , , , , ,

⎫

⎬ ⎭

A C = ⎧

⎨ ⎩ , , , ,

⎫

⎬ ⎭

I In the fourth row of the table, draw the Venn diagram for AC that includes the elements of A and U and the shaded region that represents the complement of A.

Reflect

1. Draw Conclusions Do sets always have an intersection that is not the empty set? Provide an example to support your conclusion.

Explain 1 Calculating Theoretical ProbabilitiesA probability experiment is an activity involving chance. Each repetition of the experiment is called a trial and each possible result of the experiment is termed an outcome. A set of outcomes is known as an event, and the set of all possible outcomes is called the sample space.

Probability measures how likely an event is to occur. An event that is impossible has a probability of 0, while an event that is certain has a probability of 1. All other events have a probability between 0 and 1. When all the outcomes of a probability experiment are equally likely, the theoretical probability of an event A in the sample space S is given by

P (A) = number of outcomes in the event ____ number of outcomes in the sample space

= n (A)

_ n (S)

.

2 63 74 85

See second row of table.

See third row of table.

2

1 5

1

3

2 6

4

5

3 7

6

7

4 8 9

8 9

See fourth row of table.

No. Using the example sets above, A ⋂ C = ⌀ because they do not have any elements

in common.



A2_MNLESE385900_U8M19L1 951 27/08/14 10:15 PM

COLLABORATIVE LEARNING

Small Group ActivityAsk each group to draw a spinner with 6 or 8 equal parts. Then ask them to use letters, colors, or numbers to distinguish each section of the spinner. Have them define two events based on the spinners. For example, if letters are used, the set of vowels and the set of letters in the word math. Ask students to find the probability of each event, their complements, their union, and their intersection. Have students share their work. Review which events have a probability of 1, which have a probability of 0, and why.

EXPLAIN 1 Calculating Theoretical Probabilities

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Discuss the set notation used to define theoretical probability. Connect the notation to a word description of the probability ratio, such as, the ratio of favorable outcomes in sample space to total number of outcomes in sample space.

AVOID COMMON ERRORSStudents may have difficulty identifying an event based on a union or intersection. Suggest that students draw Venn diagrams to model the experiment. They can begin by defining each set and then create the Venn diagram to show where the sets overlap.

951 Lesson 19 . 1


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Example 1 Calculate P (A) , P (A ⋃ B) , P (A ⋂ B) , and P ( A C ) for each situation.

You roll a number cube. Event A is rolling a prime number. Event B is rolling an even number.

Your grocery basket contains one bag of each of the following items: oranges, green apples, green grapes, green broccoli, white cauliflower, orange carrots, and green spinach. You are getting ready to transfer your items from your cart to the conveyer belt for check-out. Event A is picking a bag containing a vegetable first. Event B is picking a bag containing a green food first. All bags have an equal chance of being picked first.

S = ⎧ ⎨ ⎩ orange, apple, grape, broccoli, cauliflower, carrot, spinach ⎫

⎬ ⎭ , so n (S) = .

A = ⎧ ⎨ ⎩ broccoli, cauliflower, , ⎫

⎬ ⎭ , so n (A) = . So P (A) =

n ( ) _

n ( ) = _ .

A ⋃ B = ⎧ ⎨ ⎩ broccoli, , , , , grape ⎫

⎬ ⎭ , so n (A ⋃ B) = .

P (A ⋃ B) = ( )

__

( ) = _

A ⋂ B = ⎧ ⎨ ⎩ , ⎫

⎬ ⎭ , so n (A ⋂ B) =

P (A ⋂ B) = ( )

__

( ) = _

P ( ) = ( ) _

( ) = _

S = ⎧ ⎨ ⎩ 1, 2, 3, 4, 5, 6 ⎫

⎬ ⎭ , so n (S) = 6. A = ⎧

⎨ ⎩ 2, 3, 5 ⎫

⎬ ⎭ , so n (A) = 3.

So, P (A) = n (A)

_ n (S)

= 3 _ 6 = 1 _ 2 .

A ⋃ B = ⎧ ⎨ ⎩ 2, 3, 4, 5, 6 ⎫

⎬ ⎭ , so n (A ⋃ B) = 5. So, P (A ⋃ B) =

n (A ⋃ B) _

n (S) = 5 _ 6 .

A ⋂ B = ⎧ ⎨ ⎩ 2 ⎫ ⎬ ⎭ , so n (A ⋂ B) = 1. So, P (A ⋂ B) =

n (A ⋂ B) _

n (S) = 1 _ 6 .

A C = ⎧ ⎨ ⎩ 1, 4, 6 ⎫

⎬ ⎭ , so n ( A C ) = 3. So, P ( A C ) =

n ( A C ) _

n (S) = 3 _ 6 = 1 _ 2 .

carrot

carrotcauliflower

broccoli

spinach

spinach

spinach

apple

7

7

6

44

A

A

A

A c A c

∪

∩

B

B

S

S

S

S

n

n

n

n

n

n

6

2

3

2

7

7

7

Order of objects in sets may vary.


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A2_MNLESE385900_U8M19L1 952 8/26/14 8:57 PM

DIFFERENTIATE INSTRUCTION

ManipulativesEncourage students to design their own experiments to illustrate what they have learned about probability, such as calculating the complement of rolling a number with a number cube and then attempting to conform the calculation experimentally. Invite students to demonstrate their experiments before the class.

QUESTIONING STRATEGIESWhen you calculate the theoretical probabilities of events based on the same

probability experiment, can the term in the denominator of the probability ratio change? Explain. No, the term in the denominator

corresponds to the sample space, which does

not change.

If a set has no elements, what is the probability of the event represented by

the set? Explain. 0, because there are no outcomes

in the sample space that correspond to the event

If the elements of a set are the same as the elements of the sample space, what is the

probability of the event represented by the set? Explain. 1, because the number of elements in the

set is the same as the number of elements in the

sample space



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Reflect

2. Discussion In Example 1B, which is greater, P (A ⋃ B) or P (A ⋂ B) ? Do you think this result is true in general? Explain.

Your Turn

The numbers 1 through 30 are written on slips of paper that are then placed in a hat. Students draw a slip to determine the order in which they will give an oral report. Event A is being one of the first 10 students to give their report. Event B is picking a multiple of 6. If you pick first, calculate each of the indicated probabilities.

3. P (A)

4. P (A ∪ B)

5. P (A ∩ B)

6. P ( A c )

Explain 2 Using the Complement of an EventYou may have noticed in the previous examples that the probability of an event occurring and the probability of the event not occurring (i.e., the probability of the complement of the event) have a sum of 1. This relationship can be useful when it is more convenient to calculate the probability of the complement of an event than it is to calculate the probability of the event.

Probabilities of an Event and Its Complement

P (A) + P ( A c ) = 1The sum of the probability of an event and the probability of its complement is 1.

P (A) = 1 - P ( A c ) The probability of an event is 1 minus the probability of its complement.

P ( A c ) = 1 - P (A) The probability of the complement of an event is 1 minus the probability of the event.

Since P (A ∪ B) = 6 _ 7 is greater than P (A ∩ B) = 2 _ 7 , the union is more likely than the

intersection. Yes, this is generally true since the union includes all the elements from both

events, whereas the intersection contains only elements present in both sets. However,

if A = B, then the probability of the union and intersection will be the same.

P (A) = n (A)

____ n (S)

= 10 __ 30 = 1 _ 3

A ∪ B = ⎧ ⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 24, 30 ⎫

⎬ ⎭ ; P (A ∪ B) = n (A ∪ B) ______

n (S) = 14 __ 30

A ∩ B = ⎧ ⎨ ⎩ 6 ⎫

⎬ ⎭ ; P (A ∩ B) = n (A ∩ B) ______

n (S) = 1 __ 30

A c = ⎧ ⎨ ⎩ 11, 12,…, 30 ⎫

⎬ ⎭ ; P ( A c ) = n ( A c ) ____

n (S) = 20 __ 30 = 2 _ 3




LANGUAGE SUPPORT

Connect VocabularyHave students create a set of cards with diagrams to help them become familiar with the vocabulary introduced in this lesson. Help students connect the vocabulary to the notation used to represent a set, an element, the universal set, a subset, union, intersection, and complement. Have students use different colors to highlight and distinguish each relationship.

EXPLAIN 2 Using the Complement of an Event

AVOID COMMON ERRORSStudents may have difficulty understanding when to use the complement to find a probability. Point out that students may be able to find the probability directly but that the complement may provide a shortcut. Continue to remind students to create Venn diagrams to help them recognize relationships between sets.

QUESTIONING STRATEGIESWhy are there different equations that relate the probability of an event and its

complement? The three equations state the same

relationship in different ways.

953 Lesson 19 . 1


1, 1

Blue Number Cube

Whi

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umbe

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1 1, 2 1, 3 1, 4 1, 5 1, 62, 13, 14, 15, 16, 1

2, 23, 24, 25, 26, 2

2, 33, 34, 35, 36, 3

2, 43, 44, 45, 46, 4

2, 53, 54, 55, 56, 5

2, 63, 64, 65, 66, 6

2

3

4

5

6

2 3 4 5 6

4+2

5+26+27+28+2

5+36+37+38+3

4+4

5+46+47+48+4

4+5

5+56+57+58+5

4+6

5+66+67+68+6

4+345678

2 3 4 5 6Red Hearts

Blac

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Example 2 Use the complement to calculate the indicated probabilities.

You roll a blue number cube and a white number cube at the same time. What is the probability that you do not roll doubles?

Step 1 Define the events. Let A be that you do not roll doubles and A c that you do roll doubles.

Step 2 Make a diagram. A two-way table is one helpful way to identify all the possible outcomes in the sample space.

Step 3 Determine P ( A c ) . Since there are fewer outcomes for rolling doubles, it is more convenient to determine the probability of rolling doubles, which is P ( A c ) . To determine n ( A c ) , draw a loop around the outcomes in the table that correspond to A c and then calculate P ( A c ) .

P ( A c ) = n ( A c )

_ n (S)

= 6 _ 36 = 1 _ 6

Step 4 Determine P (A) . Use the relationship between the probability of an event and its complement to determine P (A) .

P (A) = 1 - P ( A c ) = 1 - 1 _ 6 = 5 _ 6

So, the probability of not rolling doubles is 5 __ 6 .

One pile of cards contains the numbers 2 through 6 in red hearts. A second pile of cards contains the numbers 4 through 8 in black spades. Each pile of cards has been randomly shuffled. If one card from each pile is chosen at the same time, what is the probability that the sum will be less than 12?

Step 1 Define the events. Let A be the event that the sum is less than 12 and A c be the event that .

Step 2 Make a diagram. Complete the table to show all the outcomes in the sample space.

Step 3 Determine P ( A c ) . Circle the outcomes in the table that correspond to A c , then determine P ( A c ) .

P ( A c ) = ⎛

⎜

⎝

⎞

⎟

⎠ _

⎛

⎜

⎝

⎞

⎟

⎠

= _

the sum is not less than 12

n A c 6

25Sn







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Step 4 Determine P (A) . Use the relationship between the probability of an event and its complement to determine P ( A c ) .

P (A) = - ( ) = - _ = _

So, the probability that the sum of the two cards is is _ .

Reflect

7. Describe a different way to calculate the probability that the sum of the two cards will be less than 12.

Your Turn

One bag of marbles contains two red, one yellow, one green, and one blue marble. Another bag contains one marble of each of the same four colors. One marble from each bag is chosen at the same time. Use the complement to calculate the indicated probabilities.

8. Probability of selecting two different colors

9. Probability of not selecting a yellow marble

Elaborate

10. Can a subset of A contain elements of A C ? Why or why not?

11. For any set A, what does A ∩ Ø equal? What does A ∪ Ø equal? Explain.

12. Essential Question Check-In How do the terms set, element, and universal set correlate to the terms used to calculate theoretical probability?

less than 12

1 125

6 19

1925

25

P A c

Use the table to count the number of outcomes in event A instead of A c , which is 19, then

divide that by the total number of outcomes to get 19 __ 25 .

AC is selecting the same color: (R1, R) , (R2, R) , (Y, Y) , (G, G) , (B, B) ;

P ( A c ) = 5 __ 20 = 1 _ 4 , so P (A) = 1 - 1 _ 4 = 3 _ 4 .

AC is selecting at least one yellow marble: (Y, R) , (Y, Y) , (Y, G) , (Y, B) , (R1, Y) , (R2, Y) , (G, Y) , (B, Y)

P ( A c ) = 8 __ 20 = 2 _ 5 , so P (A) = 1 - 2 _ 5 = 3 _ 5 .

No. The elements of a subset are contained completely within the parent set A, whereas

none of the elements of the complement of a set A are in set A by definition, and thus they

cannot be in a subset of A.

The intersection of set A and the empty set is the empty set, A ∩ Ø = Ø, since the two

sets do not have any elements in common. The union of set A and the empty set is set A,

A ∪ Ø = A, since the elements of the union are the elements in set A or the empty set.

Possible answer: To calculate probability, you need to know the number of possible

outcomes in the sample space, which is the number of elements in the universal set. You

also need to know the number of possible outcomes in the defined event, which is the

number of elements in the defined set.




ELABORATE AVOID COMMON ERRORSStudents may have trouble identifying some outcomes associated with an event. Encourage students to carefully identify all outcomes by using tables, lists, or diagrams. They can circle the outcomes of interest (often called the favorable outcomes) in the sample space.

QUESTIONING STRATEGIESHow does listing the elements in a set help you find the probability of an event associated

with the set? The probability is based on the

number of elements in the set, so you can just

count the elements for the numerator of the

probability ratio.

SUMMARIZE THE LESSONHow can you use set theory to help you calculate theoretical probabilities? You can

use the number of elements in a set to define

theoretical probability: the theoretical probability

that an event A will occur is given by P (A) = n (A) ___

n (S) ,

where S is the sample space.

955 Lesson 19 . 1


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• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

Set A is the set of factors of 12, set B is the set of even natural numbers less than 13, set C is the set of odd natural numbers less than 13, and set D is the set of even natural numbers less than 7. The universal set for these questions is the set of natural numbers less than 13.

So, A = ⎧

⎨ ⎩ 1, 2, 3, 4, 6, 12 ⎫

⎬ ⎭ , B = ⎧

⎨ ⎩ 2, 4, 6, 8, 10, 12 ⎫

⎬ ⎭ ,

C = ⎧

⎨ ⎩ 1, 3, 5, 7, 9, 11 ⎫

⎬ ⎭ , D = ⎧

⎨ ⎩ 2, 4, 6 ⎫

⎬ ⎭ , and

U = ⎧

⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ⎫

⎬ ⎭ . Answer each question.

1. Is D ⊂ A? Explain why or why not. 2. Is B ⊂ A? Explain why or why not.

3. What is A ⋂ B ? 4. What is A ⋂ C ?

5. What is A ⋃ B ? 6. What is A ⋃ C ?

7. What is A C ? 8. What is B C ?

You have a set of 10 cards numbered 1 to 10. You choose a card at random. Event A is choosing a number less than 7. Event B is choosing an odd number. Calculate the probability.

9. P (A) 10. P (B)

11. P (A ∪ B) 12. P (A ∩ B)

13. P ( A C ) 14. P ( B C )

Yes, because every element of D is also an element of A.

No, because there is at least one element of B that is not an element of A. For example, 8 is an element of B that is not an element of A.

⎧

⎨ ⎩ 2, 4, 6, 12 ⎫

⎬ ⎭ ⎧

⎨ ⎩ 1, 3 ⎫

⎬ ⎭

⎧ ⎨ ⎩ 5, 7, 9, 10, 11⎫ ⎬ ⎭ ⎧

⎨ ⎩ 1, 3, 5, 7, 9, 11 ⎫

⎬ ⎭

The sample space S = ⎧

⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ⎫

⎬ ⎭ ;

A = ⎧

⎨ ⎩ 1, 2, 3, 4, 5, 6 ⎫

⎬ ⎭

P (A) = n (A)

_ n (S)

= 6 _ 10

= 3 _ 5


⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ⎫

⎬ ⎭ ;

B = ⎧

⎨ ⎩ 1, 3, 5, 7, 9 ⎫

⎬ ⎭

P (B) = n (B)

_ n (S)

= 5 _ 10

= 1 _ 2

⎧

⎨ ⎩ 1, 2, 3, 4, 6, 8, 10, 12 ⎫

⎬ ⎭ ⎧

⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 9, 11, 12 ⎫

⎬ ⎭


⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ⎫

⎬ ⎭ ;

A ∪ B = ⎧

⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 9 ⎫

⎬ ⎭

P (A ∪ B) = n (A ∪ B)

_ n (S)

= 8 _ 10

= 4 _ 5


⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ⎫

⎬ ⎭ ;

A ∩ B = ⎧

⎨ ⎩ 1, 3, 5 ⎫

⎬ ⎭

P (A ∩ B) = n (A ∩ B)

_ n (S)

= 3 _ 10


⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ⎫

⎬ ⎭ ;

A C = ⎧

⎨ ⎩ 7, 8, 9, 10 ⎫

⎬ ⎭

P ( A C ) = n ( A C )

_ n (S)

= 4 _ 10

= 2 _ 5


⎨ ⎩ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ⎫

⎬ ⎭ ;

B C = ⎧

⎨ ⎩ 2, 4, 6, 8, 10 ⎫

⎬ ⎭

P ( B C ) = n ( B C )

_ n (S)

= 5 _ 10

= 1 _ 2




A2_MNLESE385900_U8M19L1 956 8/27/14 3:37 PMExercise Depth of Knowledge (D.O.K.) Mathematical Practices

1–8 1 Recall of Information MP.4 Modeling

9–14 1 Recall of Information MP.2 Reasoning

15–20 2 Skills/Concepts MP.2 Reasoning

21 3 Strategic Thinking MP.6 Precision

22 3 Strategic Thinking MP.4 Modeling

23–26 3 Strategic Thinking MP.3 Logic

27 3 Strategic Thinking MP.3 Logic

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreWorking with Sets

Exercises 1–8

Example 1Calculating Theoretical Probabilities

Exercises 9–14, 22, 26, 29

Example 2Using the Complement of an Event

Exercises 15–21, 23–25, 27–28

COMMUNICATING MATHDiscuss the importance of understanding the sample space. Encourage students to always list the members of the sample space before they find a probability. Discuss why this can help avoid errors, such as finding the probability of rolling a 2 with a number cube as 1 __ 5 .

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Discuss when a Venn diagram might be useful in solving a probability problem, and when another method might be easier.



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Use the complement of the event to find the probability.

15. You roll a 6-sided number cube. What is the probability that you do not roll a 2?

16. You choose a card at random from a standard deck of cards. What is the probability that you do not choose a red king?

17. You spin the spinner shown. The spinner is divided into 12 equal sectors. What is the probability of not spinning a 2?

18. A bag contains 2 red, 5 blue, and 3 green balls. A ball is chosen at random. What is the probability of not choosing a red ball?

19. Cards numbered 1–12 are placed in a bag. A ball is chosen at random. What is the probability of not choosing a number less than 5?

20. Slips of paper numbered 1–20 are folded and placed into a hat, and then a slip of paper is drawn at random. What is the probability the slip drawn has a number which is not a multiple of 4 or 5?

The probability of spinning a 2, P (2) , is 1 _ 12

.

The probability of not spinning a 2 is 1 - P (2) = 1 - 1 _ 12

= 11 _ 12

.

The probability of rolling a 2, P (2) , is 1 _ 6

.

The probability of not rolling a 2 is 1 - P (2) = 1 - 1 _ 6

= 5 _ 6

.

The probability of drawing a red

king, P (red king) , is 2 _ 52

= 1 _ 26

.

The probability of not drawing a red king

is 1 - P (red king) = 1 - 1 _ 26

= 25 _ 26

.

The probability of choosing a red ball, P (red ball) , is 2 _ 10

= 1 _ 5

.

The probability of not choosing a red ball is 1 - P (red ball) = 1 - 1 _ 5

= 4 _ 5

.

The probability of choosing a number less than 5, P (less than 5) , is 4 _ 12

= 1 _ 3

.

The probability of not choosing a number less than 5 is 1 - P (less than 5) = 1 - 1 _ 3

= 2 _ 3

.

Multiples of 4 up to 20: 4, 8, 16, 20

Multiples of 5 up to 20: 5, 10, 15, 20

The set of multiples of 4 or 5 is ⎧

⎨ ⎩ 4, 5, 8, 10, 15, 16, 20 ⎫

⎬ ⎭ .P (multiple of 4 or 5) = 7 _

20

The probability of not selecting a card that is a multiple of 4 or 5 is

1 - P (multiple of 4 or 5) = 1 - 7 _ 20

= 13 _ 20

.



A2_MNLESE385900_U8M19L1.indd 957 8/26/14 5:47 PMExercise Depth of Knowledge (D.O.K.) Mathematical Practices



INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Review the connection between likelihood and probability with students. Discuss how this can be useful when solving problems. When students calculate the probability of an event, be sure they understand what this means in the context of the original problem. For example, students should recognize that an event with a probability of 0.9 is very likely to occur, while an event with a probability of 0.1 is unlikely to occur.

AVOID COMMON ERRORSStudents may not consider the sample space when finding probabilities. Suggest that they summarize the probability ratio using words before they compute the probability.

957 Lesson 19 . 1


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21. You are going to roll two number cubes, a white number cube and a red number cube, and find the sum of the two numbers that come up.

a. What is the probability that the sum will be 6?

b. What is the probability that the sum will not be 6?

22. You have cards with the letters A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P. Event U is choosing the cards A, B, C or D. Event V is choosing a vowel. Event W is choosing a letter in the word “APPLE”. Find P (U ⋂ V ⋂ W) .

A standard deck of cards has 13 cards (2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, ace) in each of 4 suits (hearts, clubs, diamonds, spades). The hearts and diamonds cards are red. The clubs and spades cards are black. Answer each question.

23. You choose a card from a standard deck of cards at random. What is the probability that you do not choose an ace? Explain.

24. You choose a card from a standard deck of cards at random. What is the probability that you do not choose a club? Explain.

25. You choose a card from a standard deck of cards at random. Event A is choosing a red card. Event B is choosing an even number. Event C is choosing a black card. Find P (A ∩ B ∩ C) . Explain.

There are 36 possible outcomes. There are 5 ways to get a sum

of 6, where the first addend is from the white cube and the

second addend is from the red cube: 5 + 1, 4 + 2, 3 + 3, 2 + 4,

and 1 + 5. So the probability of getting a sum of 6, P (6) , is 5 __ 36 .

U ⋂ V ⋂ W = ⎧ ⎨ ⎩ A ⎫

⎬ ⎭ ; P (U ∩ V ∩ W) = 1 __ 16

The probability that the sum will not be 6 is P (not 6) = 1 - P (6) = 1 - 5 __ 36 = 31 __ 36 .

12 __ 13 ; there are 4 aces in the 52-card deck, so P (ace) = 4 __ 52 = 1 __ 13 . This means

P (not ace) = 1 - 1 __ 13 = 12 __ 13 .

3 _ 4 ; there are 13 clubs in the 52-card deck, so P (club) = 13 __ 52 = 1 _ 4 . This means

P (not club) = 1 - 1 _ 4 = 3 _ 4 .

A ∩ B ∩ C = Ø because you can never draw a card that is both red and black.

Therefore, P (A ∩ B ∩ C) = 0.





VISUAL CUESWhen students create Venn diagrams to model a sample space and sets, caution them to be sure that an element is not used more than once on the diagram. For example, have students check that a number does not appear both in Set A and in its intersection with Set B.



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26. You are selecting a card at random from a standard deck of cards. Match each event with the correct probability. Indicate a match by writing the letter of the event on the line in front of the corresponding probability.

A. Picking a card that is both red and a heart. 1 _ 52

B. Picking a card that is both a heart and an ace. 1 _ 4

C. Picking a card that is not both a heart and an ace. 51 _ 52

H.O.T. Focus on Higher Order Thinking

27. Critique Reasoning A bag contains white tiles, black tiles, and gray tiles. Someone is going to choose a tile at random. P (W) , the probability of choosing a white tile, is 1 __ 4 . A student claims that the probability of choosing a black tile, P (B) , is 3 __ 4 since P (B) = 1 - P (W) = 1 - 1 __ 4 = 3 __ 4 . Do you agree? Explain.

28. Communicate Mathematical Ideas A bag contains 5 red marbles and 10 blue marbles. You are going to choose a marble at random. Event A is choosing a red marble. Event B is choosing a blue marble. What is P (A ∩ B) ? Explain.

29. Critical Thinking Jeffery states that for a sample space S where all outcomes are equally likely, 0 ≤ P (A) ≤ 1 for any subset A of S. Create an argument that will justify his statement or state a counterexample.

No; choosing a black tile is not the complement of choosing a white tile since the bag also contains gray tiles. It is not possible to calculate P (B) from the given information.

P (red ∩ heart) = n (red ∩ heart)

__________ n (deck)

= 13 __ 52 = 1 _ 4

P (heart ∩ ace) = n (heart ∩ ace)

__________ n (deck)

= 1 __ 52

P (not (heart ∩ ace) ) = 1 - P (heart ∩ ace) = 1 - 1 __ 52 = 51 __ 52 ; the only card

that is both a heart and an ace is the ace of hearts, so there are 51 cards in

the event not (heart ∩ ace) .

B

A

C

0; A ∩ B = Ø since a marble cannot be both red and blue. So P (A ∩ B) = 0.

Assume A is a subset of S. Then 0 ≤ n (A) ≤ n (S) . For example, if S has

10 elements, the number of elements of A is greater than or equal to 0 and less

than or equal to 10. No subset of S can have fewer than 0 elements or more

than 10 elements. So 0 ≤ n (A)

___ n (S)

≤ 1. When all the outcomes are equally likely,

P (A) = n (A)

___ n (S)

. Therefore 0 ≤ P (A) ≤ 1.


DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C


JOURNALHave students write and solve their own probability problems. Remind students to use set notation in their solutions to the problems.

959 Lesson 19 . 1


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Lesson Performance Task

For the sets you’ve worked with in this lesson, membership in a set is binary: Either something belongs to the set or it doesn’t. For instance, 5 is an element of the set of odd numbers, but 6 isn’t

In 1965, Lofti Zadeh developed the idea of “fuzzy” sets to deal with sets for which membership is not binary. He defined a degree of membership that can vary from 0 to 1. For instance, a membership function mL(w) for the set L of large dogs where the degree of membership m is determined by the weight w of a dog might be defined as follows:

• A dog is a full member of the set L if it weighs 80 pounds or more. This can be written as mL(w) = 1 for w ≥ 80.

• A dog is not a member of the set L if it weighs 60 pounds or less. This can be written as mL(w) = 0 for w ≤ 60.

• A dog is a partial member of the set L if it weighs between 60 and 80 pounds. This can be written as 0 < mL(w) < 1 for 60 < w < 80.

The “large dogs” portion of the graph shown displays the membership criteria listed above. Note that the graph shows only values of m(w) that are positive.

1. Using the graph, give the approximate weights for which a dog is considered a full member, a partial member, and not a member of the set S of small dogs.

2. The union of two “fuzzy” sets A and B is given by the membership rule mA∪B (x) = maximum (mA (x) , mB (x) ) . So, for a dog of a given size, the degree of its membership in the set of small or medium-sized dogs (S ∪ M) is the greater of its degree of membership in the set of small dogs and its degree of membership in the set of medium-sized dogs.

The intersection of A and B is given by the membership rule mA⋂B (x) = minimum (mA (x) , mB (x) ) . So, for a dog of a given size, the degree of its membership in the set of dogs that are both small and medium-sized (S ⋂ M) is the lesser of its degree of membership in the set of small dogs and its degree of membership in the set of medium-sized dogs.

Using the graph above and letting S be the set of small dogs, M be the set of medium-sized dogs, and L be the set of large dogs, draw the graph of each set.

a. S ∪ M b. M ⋂ L

0 pounds to 20 pounds; between 20 pounds and 30 pounds; more than 30 pounds


DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C


EXTENSION ACTIVITY

Have students draw graphs showing fuzzy sets ranging from cold to hot. (Three sets could show cold, warm, and hot. Four sets could show cold, cool, warm, and hot. However, leave the choice of adjectives and the number of sets to students.) The vertical axis should record degrees of membership from 0 to 1. The horizontal axis should show either Fahrenheit or Celsius temperatures. Encourage students to be creative with their graphs, for example, by using colors to distinguish sets from one another. Students should write and answer at least three questions involving unions, intersections, and complements of the sets they have graphed.

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Call attention to the point in the Lesson Performance Task graph where the green medium-sized dog line and the blue big-dog line intersect. Ask students to give as much information as they can about that point. Sample answer: The point

represents a weight of around 100 pounds and a

degree of membership of around 0.3. The point

represents the highest degree of membership that a

dog of around 100 pounds can obtain

simultaneously in both the medium-sized and

big-weight categories.

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Arf has a small-dog degree of membership of x and a medium-sized dog degree of membership of y. Is x > y, x < y, or does the relationship between x and y depend on Arf ’s weight? Explain. The

relationship depends on Arf’s weight. The red and

green graphs intersect at about 40 pounds. If Arf

weighs less than 40 pounds, x > y. If Arf weighs

more than 40 pounds, x < y. If Arf weighs 40

pounds, x = y.

Scoring Rubric2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning.0 points: Student does not demonstrate understanding of the problem.



correctionkey=nl-b;ca-b correctionkey=nl-c;ca-c 19 … · explore working with sets ... number on a...

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