CorrectionKey=NL-B;CA-B 19 . 4 DO NOT EDIT--Changes must ...€¦ · Two events are mutually exclusive events if they cannot both occur in the same trial of an experiment. For example,

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Explore 1 Finding the Probability of Mutually Exclusive EventsTwo events are mutually exclusive events if they cannot both occur in the same trial of an experiment. For example, if you flip a coin it cannot land heads up and tails up in the same trial. Therefore, the events are mutually exclusive.

A number dodecahedron has 12 sides numbered 1 through 12. What is the probability that you roll the cube and the result is an even number or a 7?

A Let A be the event that you roll an even number. Let B be the event that you roll a 7. Let S be the sample space.

Complete the Venn diagram by writing all outcomes in the sample space in the appropriate region.

B Calculate P (A) .

P (A) = _ = _

C Calculate P (B) .

P (B) = _

D Calculate P (A or B) .

n (S) =

n (A or B) = n (A) + n (B)

= + =

So, P (A or B) = n (A or B)

_ n (S)

= _ .

E Calculate P (A) + P (B) . Compare the answer to Step D.

P (A) + P (B) = _ + _ = _

P (A) + P (B) P (A or B) .

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Module 19 985 Lesson 4

19.4 Mutually Exclusive and Overlapping Events

Essential Question: How are probabilities affected when events are mutually exclusive or overlapping?

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Common Core Math StandardsThe student is expected to:

S-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. Also S-CP.7

Mathematical Practices

MP.6 Precision

Language ObjectiveGive a partner an example of a mutually exclusive event. Explain how you know the events are mutually exclusive. Repeat with an example of overlapping events.

HARDCOVER PAGES 715722

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

Mutually Exclusive and Overlapping Events

ENGAGE Essential Question: How are probabilities affected when events are mutually exclusive or overlapping?The probability of mutually exclusive events is the

sum of the individual probabilities, while the

probability of overlapping events is the sum of the

individual probabilities minus the probability that

both events occur.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the photograph. Ask students to estimate the probability that two people in the photo have the same birthday. Then preview the Lesson Performance Task.

985

HARDCOVER PAGES 715722

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

Resource

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S-CP.4 For the full text of this standard, see the table starting on page CA2 of Volume 1. Also

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Explore 1 Finding the Probability of Mutually Exclusive Events

Two events are mutually exclusive events if they cannot both occur in the same trial of an

experiment. For example, if you flip a coin it cannot land heads up and tails up in the same

trial. Therefore, the events are mutually exclusive.

A number dodecahedron has 12 sides numbered 1 through 12. What is the probability that you

roll the cube and the result is an even number or a 7?

Let A be the event that you roll an even number. Let B be the event that you roll a 7.

Let S be the sample space.

Complete the Venn diagram by writing all outcomes in the sample space in the

appropriate region.

Calculate P (A) .

P (A) = _ = _

Calculate P (B) .

P (B) = _

Calculate P (A or B) .

n (S) =

n (A or B) = n (A) + n (B)

= + =

So, P (A or B) = n (A or B) _

n (S) = _ .

Calculate P (A) + P (B) . Compare the answer

to Step D.

P (A) + P (B) = _ + _ = _

P (A) + P (B) P (A or B) .

6 1

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Module 19

985

Lesson 4

19 . 4 Mutually Exclusive and

Overlapping Events

Essential Question: How are probabilities affected when events are mutually exclusive

or overlapping?

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985 Lesson 19 . 4

L E S S O N 19 . 4

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Reflect

1. Discussion How would you describe mutually exclusive events to another student in your own words? How could you use a Venn diagram to assist in your explanation?

2. Look back over the steps. What can you conjecture about the probability of the union of events that are mutually exclusive?

Explore 2 Finding the Probability of Overlapping EventsThe process used in the previous Explore can be generalized to give the formula for the probability of mutually exclusive events.

Mutually Exclusive Events

If A and B are mutually exclusive events, then P (A or B) = P (A) + P (B) .

Two events are overlapping events (or inclusive events) if they have one or more outcomes in common.

What is the probability that you roll a number dodecahedron and the result is an even number or a number greater than 7?

A Let A be the event that you roll an even number. Let B be the event that you roll a number greater than 7. Let S be the sample space.

Complete the Venn diagram by writing all outcomes in the sample space in the appropriate region.

Possible answer: Mutually exclusive events are events that have no common outcomes,

meaning that any outcome that belongs to one of the events cannot also belong to the

other event. The Venn diagram could assist in this explanation by visually showing that the

events have no outcomes in common because their intersection would be empty.

The probability of the union of events that are mutually exclusive is equal to the sum of

the probabilities of the events.





Integrate Mathematical PracticesThis lesson provides an opportunity to address Mathematical Practice MP.6, which calls for students to “communicate with precision.” In this lesson, students must decide if events are mutually exclusive or overlapping in order to apply the correct version of the Addition Rule. They may examine a variety of representations, including set notation, Venn diagrams, language analysis, and two-way tables, to make and support their decisions.

EXPLORE 1 Finding the Probability of Mutually Exclusive Events

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

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Ask students to give examples of mutually exclusive events using outcomes from different probability experiments, such as a spinner, a number cube, or a deck of cards. Discuss with students how they can be sure the events they describe are mutually exclusive.

QUESTIONING STRATEGIESExplain why, in the case of mutually exclusive events, the probability of either event is equal

to the sum of the probabilities of each event. The

events have no outcomes in common so the

probability will be the total of the individual

probabilities.

Why is a Venn diagram useful in finding the probability? The Venn diagram provides a

picture of the sets and their outcomes that shows

the sets do not overlap.

EXPLORE 2 Finding the Probability of Overlapping Events

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Ask students to give examples of overlapping events using outcomes from different probability experiments, such as a spinner, a number cube, or a deck of cards. Discuss with students how they can be sure the events they describe are overlapping.

PROFESSIONAL DEVELOPMENT

Mutually Exclusive and Overlapping Events 986


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Calculate P (A) .

P (A) = _ = _

Calculate P (B) .

P (B) = _

Calculate P (A and B) .

P (A and B) = _ = _

Use the Venn diagram to find P (A or B) .

P (A or B) = _ = _

Now, use P (A) , P (B) , and P (A and B) to calculate P (A or B) .

P (A) = P (B) = P (A and B) =

P (A) + P (B) - P (A and B) = + - =

Reflect

3. Why must you subtract P (A and B) from P (A) + P (B) to determine P (A or B) ?

4. Look back over the steps. What can you conjecture about the probability of the union of two events that are overlapping?

Explain 1 Finding a Probability From a Two-Way Table of DataThe previous Explore leads to the following rule.

The Addition Rule

P (A or B) = P (A) + P (B) - P (A and B)

Example 1 Use the given two-way tables to determine the probabilities.

P (senior or girl)

Freshman Sophomore Junior Senior TOTAL

Boy 98 104 100 94 396

Girl 102 106 96 108 412

Total 200 210 196 202 808

To determine P (senior or girl) , first calculate P (senior) , P (girl) , and P (senior and girl) .

2

16

12 12

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12

1 _ 2 5 __ 12 1 _ 4

1 _ 2 2 _ 3 1 _ 4 5 __ 12

P (A and B) must be subtracted from P (A) + P (B) to determine P (A or B) because the

outcomes in the event A and B are counted twice. Therefore, the outcomes in the

intersection must be subtracted from the total.

The probability of the union of two events that are overlapping is equal to the sum of the

probabilities of the two separate events minus the probability of both events.


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


COLLABORATIVE LEARNING

Peer-to-Peer ActivityHave students work with a partner to describe a situation in which the probability of two events is mutually exclusive. Have them formulate and answer a question about the probability. Repeat with inclusive events. Have students share their problems with the class.

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Review the set notation used for P (A or B) and P (A and B) , as well as how they are represented on Venn diagrams for both mutually exclusive and overlapping events.

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Discuss why P (A) + P (B) - P (A and B) can be used to find any probability P (A or B) . For

mutually exclusive events, P (A and B) = 0.

QUESTIONING STRATEGIESHow is P (A or B) different from P (A and B) ? Or refers to the union of the

events; and refers to the intersection, or overlap, of

the events.

How can a Venn diagram help you remember the Addition Rule for overlapping events? A

Venn diagram shows where the events overlap. This

can help me remember to subtract the intersection

so it is not counted twice.

987 Lesson 19 . 4


P (senior) = 202 _ 808 = 1 _ 4 ; P (girl) = 412 _ 808 = 103 _ 202 P (senior and girl) = 108 _ 808 = 27 _ 202

Use the addition rule to determine P (senior or girl) . P (senior or girl) = P (senior) + P (girl) - P (senior and girl)

= 1 _ 4 + 103 _ 202 - 27 _ 202

= 253 _ 404

Therefore, the probability that a student is a senior or a girl is 253 _ 404 .

B P ( (domestic or late) c ) Late On Time Total

Domestic Flights 12 108 120

International Flights 6 54 60

Total 18 162 180

To determine P ( (domestic or late) c ) , first calculate P (domestic or late) .

P (domestic) = _ = _ ; P (late) = _ = _ ; P (domestic and late) = _ = _

Use the addition rule to determine P (domestic or late) .P (domestic or late) = P (domestic) + P (late) - P (domestic and late)

= _ + _ - _ = _

Therefore, P ( (domestic or late) c ) = 1 - P (domestic or late)

= 1 - _

= _

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DIFFERENTIATE INSTRUCTION

Multiple RepresentationsDiscuss alternate ways to visualize the data in the problems presented. Have students make two-way tables, Venn diagrams, or tree diagrams as alternate ways to view the data and understand the relationships. Discuss the advantages of each representation.

EXPLAIN 1 Finding a Probability From a Two-Way Table of Data

AVOID COMMON ERRORSStudents may not use the correct total for the denominator of the probability ratio when they use a two-way table to find a probability. They may use a total from a row or column. As needed, have students extend the table to include a total that shows the sum of the columns is equal to the sum of the rows. This is the total needed for the denominator.

QUESTIONING STRATEGIESHow do you know the events in the table are inclusive? Possible answer: The overall total

of the rows and columns is the same.

How do you identify the value that overlaps to apply the Addition Rule? The overlap value is

in the cell where the column and row intersect.



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Your Turn

5. Use the table to determine P (headache or no medicine) .

Took Medicine No Medicine TOTAL

Headache 12 15 27

No Headache 48 25 73

TOTAL 60 40 100

Elaborate

6. Give an example of mutually exclusive events and an example of overlapping events.

7. Essential Question Check-In How do you determine the probability of mutually exclusive events and overlapping events?

1. A bag contains 3 blue marbles, 5 red marbles, and 4 green marbles. You choose onewithout looking. What is the probability that it is red or green?

• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

P (headache) = 27 ___ 100 ; P (no medicine) = 40 ____ 100 = 2 _ 5 ; P (headache and no medicine) = 15 ___ 100 = 3 __ 20 ;

P (headache or no medicine) = P (headache) + P (no medicine) - P (headache and

no medicine) = 27 ___ 100 + 2 _ 5 - 3 __ 20 = 13 __ 25 therefore, the probability that a person has a headache or

takes no medicine is 13 __ 25 .

Possible answer: If you roll a number cube, the event of rolling a 3 and the event of rolling

an even number are mutually exclusive because you cannot obtain both outcomes at the

same time. If you pull a card from a deck, the event of pulling an ace and the event of

pulling a spade are overlapping because you can obtain both outcomes by pulling the ace

of spades.

To determine the probability of mutually exclusive events A and B, evaluate

P (A or B) = P (A) + P (B) . To determine the probability of overlapping events A and B,

evaluate P (A or B) = P (A) + P (B) - P (A and B) .

Let S be the sample space, A be the event that you choose a red marble, and B be the

event that you choose a green marble. n (S) = 12, n (A or B) = n (A) + n (B) = 5 + 4 = 9;

P (A or B) = n (A or B)

______ n (S)

= 9 __ 12 = 3 _ 4 ; the probability that you choose a red marble or a green

marble is 3 _ 4 .


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ELABORATE AVOID COMMON ERRORSStudents may forget to subtract the overlapping probability when finding the probability of overlapping events. Encourage students to draw Venn diagrams to help them remember how to apply the Addition Rule to solve probability problems.

SUMMARIZE THE LESSONHow do you find the probability of mutually exclusive events and overlapping events? For

mutually exclusive events A and B,

P (A or B) = P (A) + P (B) . For overlapping events A

and B, P (A or B) = P (A) + P (B) - P (A and B) .

LANGUAGE SUPPORT

Connect VocabularyFor some students, the phrase overlapping events may be unclear. Separate the class into groups. Have students work together to create lists of overlapping events other than those in the examples from the lesson.

989 Lesson 19 . 4


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2. A number icosahedron has 20 sides numbered 1 through 20. What is the probability that the result of a roll is a number less than 4 or greater than 11?

3. A bag contains 26 tiles, each with a different letter of the alphabet written on it. You choose a tile without looking. What is the probability that you choose a vowel (a, e, i, o, or u) or a letter in the word GEOMETRY?

4. Persevere in Problem Solving You roll two number cubes at the same time. Each cube has sides numbered 1 through 6. What is the probability that the sum of the numbers rolled is even or greater than 9? (Hint: Create and fill out a probability chart.)

Let S be the sample space, A be the event that you roll a number less than 4, and B be the event that you roll a number greater than 11.

n (S) = 20, n (A or B) = n (A) + n (B) = 3 + 9 = 12


______ n (S)

= 12 __ 20 = 3 _ 5

The probability that the result is a number less than 4 or greater than 11 is 3 _ 5 .

Let S be the sample space, A be the event that you choose a vowel, and B be the event that you choose a letter in the word GEOMETRY.

n (S) = 26, n (A or B) = n (A) + n (B) - n (A and B) = 5 + 7 - 2 = 10


______ n (S)

= 10 __ 26 = 5 __ 13

The probability that you choose a vowel or a letter in the word GEOMETRY is 5 __ 13 .

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1 1 + 1 1 + 2 1 + 3 1 + 4 1 + 5 1 + 6

2 2 + 1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6

3 3 + 1 3 + 2 3 + 3 3 + 4 3 + 5 3 + 6

4 4 + 1 4 + 2 4 + 3 4 + 4 4 + 5 4 + 6

5 5 + 1 5 + 2 5 + 3 5 + 4 5 + 5 5 + 6

6 6 + 1 6 + 2 6 + 3 6 + 4 6 + 5 6 + 6

Let S be the sample space, A be the event that the sum of the numbers is even, and B be the event that the sum of the numbers is greater than 9.

n (S) = 36, n (A or B) = n (A) + n (B) - n (A and B) = 18 + 6 - 4 = 20


______ n (S)

= 20 __ 36 = 5 _ 9

The probability that the sum of the numbers rolled is even or greater than 9 is 5 _ 9 .




A2_MNLESE385900_U8M19L4.indd 990 26/08/14 9:42 PMExercise Depth of Knowledge (D.O.K.) Mathematical Practices

1–15 2 Skills/Concepts MP.2 Reasoning

16 2 Skills/Concepts MP.3 Logic

17–21 2 Skills/Concepts MP.1 Problem Solving

22 3 Strategic Thinking MP.3 Logic


24 3 Strategic Thinking MP.4 Modeling


EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

Explore 1Finding the Probability of Mutually Exclusive Events

Exercises 1–2, 22–23

Explore 2Finding the Probability of Overlapping Events

Exercises 3–4

Example 1Finding a Probability From a Two-Way Table of Data

Exercises 5–10, 11–21

COMMUNICATING MATHDiscuss the importance of filling in the total values for each row and column when the total values for a two-way table are not given.



The table shows the data for car insurance quotes for 125 drivers made by an insurance company in one week.

Teen Adult (20 or over) Total

0 accidents 15 53 68

1 accident 4 32 36

2+ accidents 9 12 21

Total 28 97 125

You randomly choose one of the drivers. Find the probability of each event.

5. The driver is an adult. 6. The driver is a teen with 0 or 1 accident.

7. The driver is a teen. 8. The driver has 2+ accidents.

9. The driver is a teen and has 2+ accidents.

10. The driver is a teen or a driver with 2+ accidents.

Use the following information for Exercises 11–16. The table shown shows the results of a customer satisfaction survey for a cellular service provider, by location of the customer. In the survey, customers were asked whether they would recommend a plan with the provider to a friend.

Arlington Towson Parkville Total

Yes 40 35 41 116

No 18 10 6 34

Total 58 45 47 150

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Total Adults _________ Total Drivers

= 97 ___ 125

Total Teens _________ Total Drivers

= 28 ___ 125

Teen with 0 or 1 accident ________________ Total Drivers

= 19 ___ 125

Drivers with 2+ accidents _________________ Total Drivers

= 21 ___ 125

Teens with 2+ accidents ________________ Total Drivers

= 9 ___ 125

Teen or Driver with 2+ accidents _____________________ Total Drivers

= 28 ___ 125 + 21 ___ 125 - 9 ___ 125 = 40 ___ 125 = 8 __ 25




INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Discuss with students why it is easier not to simplify the fractions until after they have used the Addition Rule to calculate probabilities.

AVOID COMMON ERRORSStudents may not recognize overlapping events or may forget to subtract the overlap. Suggest that students first decide if the events can overlap. If so, students should identify the probability of this event first. Discuss how students can use and to help them recognize whether events overlap.

991 Lesson 19 . 4


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One of the customers that was surveyed was chosen at random. Find the probability of each event.

11. The customer was from Towson and said No. 12. The customer was from Parkville.

13. The customer said Yes. 14. The customer was from Parkville and said Yes.

15. The customer was from Parkville or said Yes.

16. Explain why you cannot use the rule P (A or B) = P (A) + P (B) in Exercise 15.

Use the following information for Exercises 17–21. Roberto is the owner of a car dealership. He is assessing the success rate of his top three salespeople in order to offer one of them a promotion. Over two months, for each attempted sale, he records whether the salesperson made a successful sale or not. The results are shown in the chart.

Successful Unsuccessful Total

Becky 6 6 12

Raul 4 5 9

Darrell 6 9 15

Total 16 20 36

Roberto randomly chooses one of the attempted sales.

17. Find the probability that the sale was one of Becky’s or Raul’s successful sales.

Towson and said no _____________ Total

= 10 ___ 150 = 1 __ 15 Parkville ______ Total

= 47 ___ 150

Yes ____ Total

= 116 ___ 150 = 58 __ 75 Parkville and said Yes ______________ Total

= 41 ___ 150

Parkville or said Yes _____________ Total

= 47 ___ 150 + 116 ___ 150 - 41 ___ 150 = 122 ___ 150 = 61 __ 75

The events are not mutually exclusive because there are 41 customers who both live in Parkville and said yes, and they get counted twice when adding P (A) and P (B) , so you must use the more general rule P (A or B) = P (A) + P (B) – P (A and B) .

Let A be the set of Becky’s successful sales attempts, let B be the set of Raul’s successful sales attempts, and let S be the set of all sales attempts.

n (S) = 36

n (A ∪ B) = n (A) + n (B) = 6 + 4 = 10

P (A ∪ B) = n (A ∪ B)

______ n (S)

= 10 __ 36 = 5 __ 18

The probability that the sale was one of Becky’s or Raul’s successful sales is 5 __ 18 .





AVOID COMMON ERRORSStudents might treat inclusive events as mutually exclusive and double count a probability. A total probability greater than 1 could indicate this error. Remind students to consider whether the events can occur simultaneously before they choose the form of the addition rule to use.



18. Find the probability that the sale was one of the unsuccessful sales or one of Raul’s successful sales.

19. Find the probability that the sale was one of Darrell’s unsuccessful sales or one of Raul’s unsuccessful sales.

20. Find the probability that the sale was an unsuccessful sale or one of Becky’s attempted sales.

21. Find the probability that the sale was a successful sale or one of Raul’s attempted sales.

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Let A be the set of all unsuccessful sales attempts, let B be the set of Raul’s successful sales attempts, and let S be the set of all sales attempts.

n (S) = 36

n (A ∪ B) = n (A) + n (B) = 4 + 20 = 24

P (A ∪ B) = n (A ∪ B)

______ n (S)

= 24 __ 36 = 2 _ 3

The probability that the sale was one of the unsuccessful sales or one of Raul’s

successful sales is 2 _ 3 .

Let A be the set of Darrell’s unsuccessful sales attempts, let B be the set of Raul’s unsuccessful sales attempts, and let S be the set of all sales attempts.

n (S) = 36

n (A ∪ B) = n (A) + n (B) = 9 + 5 = 14

P (A ∪ B) = n (A ∪ B)

______ n (S)

= 14 __ 36 = 7 __ 18

The probability that the sale was one of Darrell’s unsuccessful sales or one of Raul’s

unsuccessful sales is 7 __ 18 .

Let A be the set of all unsuccessful sales attempts, let B be the set of Becky’s sales attempts, and let S be the set of all sales attempts.

n (S) = 36

n (A ∪ B) = n (A) + n (B) - n (A ∩ B) = 20 + 12 - 6 = 26

P (A ∪ B) = n (A ∪

B) ______

n (S) = 26 __ 36 = 13 __ 18

The probability that the sale was an unsuccessful sale or one of Becky’s attempted

sales is 13 __ 18 .

Let A be the set of all successful sales attempts, let B be the set of Raul’s sales attempts, and let S be the set of all sales attempts.

n (S) = 36

n (A ∪ B) = n (A) + n (B) - n (A ∩ B) = 16 + 9 - 4 = 21

P (A ∪ B) = n (A ∪ B)

______ n (S)

= 21 __ 36 = 7 __ 12

The probability that the sale was a successful sale or one of Raul’s attempted sales is 7 __ 12 .




993 Lesson 19 . 4


22. You are going to draw one card at random from a standard deck of cards. 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. Which of the following have a probability of less than 1 __ 4 ? Choose all that apply.

a. Drawing a card that is a spade and an ace

b. Drawing a card that is a club or an ace

c. Drawing a card that is a face card or a club

d. Drawing a card that is black and a heart

e. Drawing a red card and a number card from 2–9

H.O.T. Focus on Higher Order Thinking

23. Draw Conclusions A survey of 1108 employees at a software company finds that 621 employees take a bus to work and 445 employees take a train to work. Some employees take both a bus and a train, and 321 employees take only a train. To the nearest percent, find the probability that a randomly chosen employee takes a bus or a train to work. Explain.

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a. P (spade ∩ ace) = n (spade ∩ ace)

__________ n (deck)

= 1 __ 52 < 1 _ 4

b. P (club ∪ ace) = n (club ∪ ace) _________

n (deck) = n (club) + n (ace) - n (club ∩ ace)

______________________ n (deck)

= 13 + 4 - 1 _______ 52 = 16 __ 52 = 4 __ 13 > 1 _ 4

c. P (face ∪ club) = n (face ∪ club) __________

n (deck) = n (face) + n (club) - n (face ∩ club)

_______________________ n (deck)

= 12 + 13 - 3 ________ 52 = 22 __ 52

= 11 __ 26 > 1 _ 4

d. P (black ∩ heart) = n (black ∩ heart) ___________

n (deck) = 0 __ 52 = 0 < 1 _ 4

e. P (red ∩ 2 - 9) = n (red ∩ 2 - 9) __________

n (deck) = 16 __ 52 = 4 __ 13 > 1 _ 4

If 321 employees take only a train, and 445 total employees take a train, then

445 - 321 = 124 people take both a bus and a train to work.

n (A ∪ B) = n (bus) + n (train) - n (bus and train) = 621 + 445 -124 = 942

n (S) = n (total surveyed) = 1108

P (A ∪ B) = n (A ∪ B)

______ n (S)

= 942 ____ 1108 = 471 ___ 554 ≈ 85%

The probability that a randomly chosen employee takes a bus or a train to work is 85%.







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24. Communicate Mathematical Ideas Explain how to use a Venn diagram to find the probability of randomly choosing a multiple of 3 or a multiple of 4 from the set of numbers from 1 to 25. Then find the probability.

25. Explain the Error Sanderson attempted to find the probability of randomly choosing a 10 or a diamond from a standard deck of playing cards. He used the following logic:

Let S be the sample space, A be the event that the card is a 10, and B be the event that the card is a diamond.

There are 52 cards in the deck, so n (S) = 52.

There are four 10s in the deck, so n (A) = 4.

There are 13 diamonds in the deck, so n (B) = 13.

One 10 is a diamond, so n (A ∩ B) = 1.

P (A ∪ B) = n (A ∪ B)

_ n (S)

= n (A) · n (B) - n (A ∩ B) ___

n (S) = 4 · 13 - 1 _ 52 = 51 _ 52

Describe and correct Sanderson’s mistake.

A BA ⋂ B S2 1 7 25 5 10

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Let A be the set of multiples of 3 from 1 to 25 and B be the set of multiples of 4 from 1 to 25. Create a Venn diagram representing the sets A and B.

Add the numbers of elements in A and B, and then subtract the number of elements in the overlap to get the numerator of the probability. The denominator is 25.

n (A ∪ B) = 8 + 6 - 2 = 12

P (A ∪ B) = 12 __ 25

The probability of randomly choosing a multiple of 3 or a multiple of 4 from the set of

numbers from 1 to 25 is 12 __ 25 .

When finding n (A ∪ B) , n (A) should be added to n (B) , not multiplied.

P (A ∪ B) = n (A) + n (B) - n (A ∩ B)

_______________ n (S)

= 4 + 13 - 1 _______ 52 = 16 ___ 52 = 4 __ 13




JOURNALHave students explain how the probabilities of mutually exclusive and inclusive events are really two versions of the same problem.

995 Lesson 19 . 4


Lesson Performance Task

What is the smallest number of randomly chosen people that are needed in order for there to be a better than 50% probability that at least two of them will have the same birthday? The astonishing answer is 23. Follow these steps to find why.

1. Can a person have a birthday on two different days? Use the vocabulary of this lesson to explain your answer.

Looking for the probability that two or more people in a group of 23 have matching birthdays is a challenge. Maybe there is one match but maybe there are five matches or seven or fourteen. A much easier way is to look for the probability that there are no matches in a group of 23. In other words, all 23 have different birthdays. Then use that number to find the answer.

2. There are 365 days in a non-leap year.

a. Write an expression for the number of ways can you assign different birthdays to 23 people. (Hint: Think of the people as standing in a line, and you are going to assign a different number from 1 to 365 to each person.)

b. Write an expression for the number ways can you assign any birthday to 23 people. (Hint: Now think about assigning any number from 1 to 365 to each of 23 people.)

c. How can you use your answers to (a) and (b) to find the probability that no people in a group of 23 have the same birthday? Use a calculator to find the probability to the nearest ten-thousandth.

d. What is the probability that at least two people in a group of 23 have the same birthday? Explain your reasoning.

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1. No; The set of days of the year when you have a birthday and the set of days of the year when you do not have a birthday are mutually exclusive, not overlapping.

2. a. Because assigned numbers must be different, find the number of permutations of 365 numbers taken 23 at a time: 365 P 23

b. Because assigned numbers can be the same, the number of choices for each person is 365, so form a product where 365 appears as a factor 23 times: 3 65 23

c. Divide 365 P 23 by 3 65 23 and get 0.4927 to the nearest ten-thousandth.

d. The complement of the event that at least two people in a group of 23 have the same birthday is the event that no people in a group of 23 have the same birthday, so P(at least two people in a group of 23 have the same birthday) = 1 - P(no people in a group of 23 have the same birthday) ≈ 1 - 0.4927 = 0.5073 or 50.73%.




EXTENSION ACTIVITY

Present another surprising probability paradox: (1) Mister Jones has two children. The oldest is a girl. What is the probability that the other child is a girl? Explain your reasoning. 1 _

2 ; 2 possible outcomes for second child: {B, G}; 1 favorable

outcome: G (2) Mister Smith has two children. At least one of them is a girl. What is the probability that the other child is a girl? Explain your reasoning. 1 _

3 ; 3

possible combinations: {GG, BG, GB}; 1 favorable outcome: GG It may seem counterintuitive that specifying “at least” changes the probability that the second child will be a girl. To test this, have students flip two coins (H = girl, T = boy). They should ignore TT (boy, boy). If one is a heads, they should count the number of times the other is also a heads.

AVOID COMMON ERRORSStudents hearing the birthday paradox for the first time often misunderstand it as claiming that in a group of 23 randomly chosen people, the chances are better than 50–50 that one of them will have a specific given birthday, say July 23. Students are correct in thinking that that is highly unlikely. Stress that the paradox is that in a random group of 23, the chances are better than 50–50 that two of them will have the same birthday. The actual date of the birthday, however, is unstated.

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Tammy’s birthday is July 23. How large must a group of randomly chosen people be in order for there to be a better than 50% chance that one of them will have the same birthday she has? 183; 183

_ 365

≈ 50.1%

Scoring Rubric2 points: The student’s answer is an accurate and complete execution of the task or tasks.1 point: The student’s answer contains attributes of an appropriate response but is flawed.0 points: The student’s answer contains no attributes of an appropriate response.



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CorrectionKey=NL-B;CA-B 19 . 4 DO NOT EDIT--Changes must ...€¦ · Two events are mutually exclusive events if they cannot both occur in the same trial of an experiment. For example,