Common Core State Standards 1 Interactive Learning CC-13 ...terra-dade.enschool.org/ourpages/auto/2015/9/3/57271909/Alg II CC … · 03/09/2015 · an expression involving probabilities

44 Chapter 12 Data Analysis and Probability

Probability of Compound EventsObjectives To find probabilities of mutually exclusive and overlapping events

To find probabilities of independent and dependent events

•compound event•mutually

exclusive events•overlapping

events•independent

events•dependent

events

LessonVocabulary

The portable music player at the right is set to choose a song at random from the playlist. What is the probability that the next song played is a rock song by an artist whose name begins with the letter A? How did you find your answer?

Artist Category SongsAbsolute Value Rock 10Algebras Pop 12Arithmetics Rock 6FOILs Pop 5Pascal’s Triangle Country 12Pi Rock 11

Key Concept Probability of A or B

Probability of Mutually Exclusive EventsIf A and B are mutually exclusive events, P (A or B) = P(A) + P(B).

Probability of Overlapping EventsIf A and B are overlapping events, P (A or B) = P (A) + P (B) - P (A and B).

In the Solve It, you found the probability that the next song is both a rock song and also a song by an artist whose name begins with the letter A. This is an example of a compound event, which consists of two or more events linked by the word and or the word or.

Essential Understanding You can write the probability of a compound event as an expression involving probabilities of simpler events. This may make the compound probability easier to find.

When two events have no outcomes in common, the events are mutually exclusive events. If A and B are mutually exclusive events, then P(A and B) = 0. When events have at least one outcome in common, they are overlapping events.

You need to determine whether two events A and B are mutually exclusive before you can find P (A or B).

Start with a plan. How many songs are there?

CC-13 MACC.912.S-CP.2.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. Also MACC.912.S-CP.2.8

MP 1, MP 2, MP 3, MP 4, MP 6

Common Core State Standards

MATHEMATICAL PRACTICES

HSM15_A2Reg_SE_CC_13_TrKit.indd Page 44 02/08/13 8:38 PM gg-018 /120/PE01457/TRANSITION_KITS/NA/ANCILLARY/2015/XXXXXXXXXX/Layout/Interior_Files/A ...

Problem 1

Got It?

Lesson 12-8 Probability of Compound Events 45

Key Concept Probability of Two Independent Events

If A and B are independent events, P (A and B) = P(A) # P(B).

Mutually Exclusive and Overlapping Events

Suppose you spin a spinner that has 20 equal-sized sections numbered from 1 to 20.

A What is the probability that you spin a 2 or a 5?

Because the spinner cannot land on both 2 and 5, the events are mutually exclusive.

P (2 or 5) = P (2) + P (5)

= 120 + 1

20 Substitute.

= 220 = 1

10 Simplify.

The probability that you spin a 2 or a 5 is 110.

B What is the probability that you spin a number that is a multiple of 2 or 5?

Since a number can be a multiple of 2 and a multiple of 5, such as 10, the events are overlapping.

P (multiple of 2 or multiple of 5)

= P (multiple of 2) + P (multiple of 5) - P (multiple of 2 and 5)

= 1020 + 4

20 - 220 Substitute.

= 1220 = 3

5 Simplify.

The probability that you spin a number that is a multiple of 2 or a multiple of 5 is 35.

1. Suppose you roll a standard number cube. a. What is the probability that you roll an even number or a number less

than 4? b. What is the probability that you roll a 2 or an odd number?

A standard set of checkers has equal numbers of red and black checkers. The diagram at the right shows the possible outcomes when randomly choosing a checker, putting it back, and choosing again. The probability of getting a red on either choice is 12. The first choice, or event, does not affect the second event. The events are independent.

Two events are independent events if the occurrence of one event does not affect the probability of the second event.

RedRedBlack

Black

1st Choice 2nd Choice

RedBlack

hsm11a1se_1208_t08495.ai

How many multiples are there?There are 10 multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. There are 4 multiples of 5: 5, 10, 15, and 20. There are 2 multiples of 2 and 5: 10 and 20.

HSM15_A2Reg_SE_CC_13_TrKit.indd Page 45 02/08/13 8:38 PM gg-018 /120/PE01457/TRANSITION_KITS/NA/ANCILLARY/2015/XXXXXXXXXX/Layout/Interior_Files/A ...44 Common Core

HSM15_A2Reg_SE_TrKit.indd Page 44 03/08/13 3:52 PM gg-018 /120/PE01457/TRANSITION_KITS/NA/ANCILLARY/2015/XXXXXXXXXX/Layout/Interior_Files/A ...

44 Common Core

FACILITATEq How many songs on the music device are rock songs?

How many are performed by an artist whose name begins with the letter A? [27; 28]

q How many songs on the music device are rock songs that are performed by an artist whose name begins with the letter A? [16]

1 Interactive LearningSolve It!PURPOSE To determine the probability of a compound event using simple probabilityPROCESS Students may use simple probability by determining the number of favorable outcomes and comparing it to the number of possible outcomes.

events are said to be independent if the probability of one event occurring is not dependent on the other event occurring.Have students look at compound probability problems by first determining whether the first event affects the second event. Once students determine whether the events are dependent or independent, they can select the appropriate equation. It may sometimes be difficult to determine whether events are independent, but it is crucial mathematicaly: P (A and B) =P (A) # P (B) if and only if A and B are independent events.

Mathematical PracticeAttend to precision. Students will make explicit use of the terms “mutually exclusive events” and “overlapping events” and will determine when to apply each.

BIG idea ProbabilityESSEntial UndERStandingS•Theprobabilityofacompoundeventcan

sometimes be found from expressions of the probabilities of simpler events.

•Differentmethodsmustbeusedforfindingtheprobability of two dependent events compared to finding the probability of two independent events.

Math BackgroundA compound event in the study of probability is an event that consists of two or more simple probability events. When two simple events constitute a compound event, the two events can be either a union in which one or the other event occurs, or an intersection in which both of the events occur. The two events are said to be mutually exclusive if the probability of both events occurring is zero. The two

Preparing to Teach

anSWER See Solve It in Answers on next page.COnnECt tHE MatH Students explore a compound event in the Solve It. In the lesson, students will learn about compound events, mutually exclusive events, overlapping events, independent events, and dependent events and how the type of event affects the probability of the event.

2 Guided InstructionTake NoteUse Venn diagrams and the data provided in the Solve It to illustrate the concepts of mutually exclusive and overlapping events.

CC-13


Probability of Compound EventsObjectives To find probabilities of mutually exclusive and overlapping events

To find probabilities of independent and dependent events

•compound event•mutually

exclusive events•overlapping

events•independent

events•dependent

events

LessonVocabulary

The portable music player at the right is set to choose a song at random from the playlist. What is the probability that the next song played is a rock song by an artist whose name begins with the letter A? How did you find your answer?

Artist Category SongsAbsolute Value Rock 10Algebras Pop 12Arithmetics Rock 6FOILs Pop 5Pascal’s Triangle Country 12Pi Rock 11

Key Concept Probability of A or B

Probability of Mutually Exclusive EventsIf A and B are mutually exclusive events, P (A or B) = P(A) + P(B).

Probability of Overlapping EventsIf A and B are overlapping events, P (A or B) = P (A) + P (B) - P (A and B).

In the Solve It, you found the probability that the next song is both a rock song and also a song by an artist whose name begins with the letter A. This is an example of a compound event, which consists of two or more events linked by the word and or the word or.

Essential Understanding You can write the probability of a compound event as an expression involving probabilities of simpler events. This may make the compound probability easier to find.

When two events have no outcomes in common, the events are mutually exclusive events. If A and B are mutually exclusive events, then P(A and B) = 0. When events have at least one outcome in common, they are overlapping events.

You need to determine whether two events A and B are mutually exclusive before you can find P (A or B).

Start with a plan. How many songs are there?

CC-13 MACC.912.S-CP.2.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. Also MACC.912.S-CP.2.8

MP 1, MP 2, MP 3, MP 4, MP 6

Common Core State Standards



Problem 1

Got It?


Key Concept Probability of Two Independent Events

If A and B are independent events, P (A and B) = P(A) # P(B).

Mutually Exclusive and Overlapping Events

Suppose you spin a spinner that has 20 equal-sized sections numbered from 1 to 20.

A What is the probability that you spin a 2 or a 5?

Because the spinner cannot land on both 2 and 5, the events are mutually exclusive.

P (2 or 5) = P (2) + P (5)

= 120 + 1

20 Substitute.

= 220 = 1

10 Simplify.

The probability that you spin a 2 or a 5 is 110.

B What is the probability that you spin a number that is a multiple of 2 or 5?

Since a number can be a multiple of 2 and a multiple of 5, such as 10, the events are overlapping.

P (multiple of 2 or multiple of 5)

= P (multiple of 2) + P (multiple of 5) - P (multiple of 2 and 5)

= 1020 + 4

20 - 220 Substitute.

= 1220 = 3

5 Simplify.

The probability that you spin a number that is a multiple of 2 or a multiple of 5 is 35.

1. Suppose you roll a standard number cube. a. What is the probability that you roll an even number or a number less

than 4? b. What is the probability that you roll a 2 or an odd number?

A standard set of checkers has equal numbers of red and black checkers. The diagram at the right shows the possible outcomes when randomly choosing a checker, putting it back, and choosing again. The probability of getting a red on either choice is 12. The first choice, or event, does not affect the second event. The events are independent.

Two events are independent events if the occurrence of one event does not affect the probability of the second event.

RedRedBlack

Black

1st Choice 2nd Choice

RedBlack

hsm11a1se_1208_t08495.ai

How many multiples are there?There are 10 multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. There are 4 multiples of 5: 5, 10, 15, and 20. There are 2 multiples of 2 and 5: 10 and 20.

HSM15_A2Reg_SE_CC_13_TrKit.indd Page 45 02/08/13 8:38 PM gg-018 /120/PE01457/TRANSITION_KITS/NA/ANCILLARY/2015/XXXXXXXXXX/Layout/Interior_Files/A ...CC-13 Probability of Compound Events 45


CC-13 45

AnswersSolve It!27; explanations may vary.

Got It? 1. a. 5

6

b. 23

q Is it possible for the spinner to land on both 2 and 5 during the same spin? Explain. [No; there is only one number on each section.]

q If you used the formula for overlapping events to determine the probability in 1A, would you still arrive at the same answer? [Yes; the probability of landing on both 2 and 5 is zero, so you would get the same answer.]

q Is it possible to land on both a multiple of 2 and a multiple of 5? Explain. [Yes; the numbers 10 and 20 are multiples of both 2 and 5.]

q If you used the formula for mutually exclusive events to determine the probability in 1B, would you still arrive at the same answer? [No, because you would count some of the sections twice as favorable outcomes.]

Got It?

Problem 1

q Which formula should you use to compute the probability in 1a? 1b? [formula for overlapping events; formula for mutually exclusive events]

Take NoteAsk students to state several more examples and a nonexample of independent events.

Problem 2

Problem 3

Got It?

Got It?


Finding the Probability of Independent Events

Suppose you roll a red number cube and a blue number cube. What is the probability that you will roll a 3 on the red cube and an even number on the blue cube?

P (red 3) = 16 Only one of the six numbers is a 3.

P (blue even) = 36 = 1

2 Three of the six numbers are even.

P (red 3 and blue even) = P (red 3) # P (blue even)

= 16# 1

2 = 112 Substitute and then simplify.

The probability is 112.

2. You roll a red number cube and a blue number cube. What is the probability that you roll a 5 on the red cube and a 1 or 2 on the blue cube?

Selecting With Replacement

Games You choose a tile at random from the game tiles shown. You replace the first tile and then choose again. What is the probability that you choose a dotted tile and then a dragon tile?

Because you replace the first tile, the events are independent.

P (dotted) = 415 4 of the 15 tiles are dotted.

P (dragon) = 315 = 1

5 3 of the 15 tiles are dragons.

P (dotted and dragon) = P (dotted) # P (dragon)

= 415

# 15 Substitute.

= 475 Simplify.

The probability that you will choose a dotted tile and then a dragon tile is 4

75.

3. In Problem 3, what is the probability that you randomly choose a bird and then, after replacing the first tile, a flower?

Two events are dependent events if the occurrence of one event affects the probability of the second event. For example, suppose in Problem 3 that you do not replace the first tile before choosing another. This changes the set of possible outcomes for your second selection.

Why are the events independent when you select with replacement?When you replace the tile, the conditions for the second selection are exactly the same as for the first selection.

Are the events independent?Yes. The outcome of rolling one number cube does not affect the outcome of rolling another number cube.


Problem 4

Got It?

Problem 5


Selecting Without Replacement

Games Suppose you choose a tile at random from the tiles shown in Problem 3. Without replacing the first tile, you select a second tile. What is the probability that you choose a dotted tile and then a dragon tile?

Because you do not replace the first tile, the events are dependent.


P (dragon after dotted) = 314 3 of the 14 remaining tiles are dragons.

P (dotted then dragon) = P (dotted) # P (dragon after dotted)

= 415

# 314 = 2

35 Substitute and then simplify.

The probability that you will choose a dotted tile and then a dragon tile is 235.

4. In Problem 4, what is the probability that you will randomly choose a flower and then, without replacing the first tile, a bird?

Finding the Probability of a Compound Event

Essay Contest One freshman, 2 sophomores, 4 juniors, and 5 seniors receive top scores in a school essay contest. To choose which 2 students will read their essays at the town fair, 2 names are chosen at random from a hat. What is the probability that a junior and then a senior are chosen?

The first outcome affects the probability of the second. So the events are dependent.

P (junior) = 412 = 1

3 4 of the 12 students are juniors.

P (senior after junior) = 511 5 of the 11 remaining students are seniors.

P (junior then senior) = P (junior) # P (senior after junior)

= 13# 5


The probability that a junior and then a senior are chosen is 533.

hsm11a1se_1208_t08505

9876543210

�

9876543210

9876543210

9876543210

9876543210

9876543210

2 / 53

Determine whether the events are dependent or independent and use the formula that applies.

P (junior then senior)Grade levels of the 12 students

Key Concept Probability of Two Dependent Events

If A and B are dependent events, P (A then B) = P (A) # P (B after A).

How is P(dragon after dotted) different from P(dragon)?After selecting the first tile without replacement, there is one less tile to choose from for the second choice.



46 Common Core

q What is the total number of outcomes possible when the two number cubes are rolled simultaneously? Explain. [Using the Multiplication Counting Principle, there are 6 # 6 = 36 possible outcomes.]

q What is the total number of favorable outcomes possible when the two number cubes are rolled simultaneously? [Using the Multiplication Counting Principle, there are 1 # 3 = 3 possible favorable outcomes.]

1. A dartboard has 12 equally sized sections numbered from 1 to 12.

a. What is the probability of throwing a dart that lands on an odd number?

b. What is the probability of throwing a dart that lands on a multiple of 3?

anSWER a. 12 b. 13 2. Suppose you roll a number cube

and flip a coin. What is the probability of rolling a number greater than 2 and flipping heads?

anSWER 13

Additional Problems 3. A bag contains 4 red chips,

3 green chips, 6 blue chips, and 5 black chips. Andrew selects a chip at random. He replaces the chip and then selects another one at random. What is the probability that he selects a red chip, then a black chip?

anSWER 581

4. Refer to the information given in Additional Problem 3. Suppose Andrew selects a chip at random, does not replace it, then selects another chip at random. What is the probability that he selects a blue chip, then a green chip?

anSWER 117

5. Justin has 8 rock songs, 3 hip hop songs, 5 classical music songs, and 4 country songs in a playlist on his mp3 player. Suppose he plays songs at random from the playlist. If the mp3 player will not play the same song twice in a row, what is the probability that he will hear a rock song followed by a country song?

anSWER 895

Problem 2The probability can be calculated by interpreting this event as a simple event.

Got It?Ask students to describe an event involving the number cubes that has a probability of 12.

q What is the probability of choosing a bird tile? a flower tile? [ 2

15; 115]

Problem 3Show students that the probability can be computed using the Multiplication Counting Principle. The number of ways to choose a dotted tile first and then a dragon is 4 # 3 = 12 ways. The number of ways to choose two tiles is 15 # 15 = 225. Therefore, the probability is 12

225 = 475.

Got It?

Problem 2

Problem 3

Got It?

Got It?


Finding the Probability of Independent Events

Suppose you roll a red number cube and a blue number cube. What is the probability that you will roll a 3 on the red cube and an even number on the blue cube?

P (red 3) = 16 Only one of the six numbers is a 3.

P (blue even) = 36 = 1

2 Three of the six numbers are even.

P (red 3 and blue even) = P (red 3) # P (blue even)

= 16# 1


The probability is 112.

2. You roll a red number cube and a blue number cube. What is the probability that you roll a 5 on the red cube and a 1 or 2 on the blue cube?

Selecting With Replacement

Games You choose a tile at random from the game tiles shown. You replace the first tile and then choose again. What is the probability that you choose a dotted tile and then a dragon tile?

Because you replace the first tile, the events are independent.


P (dragon) = 315 = 1

5 3 of the 15 tiles are dragons.

P (dotted and dragon) = P (dotted) # P (dragon)

= 415

# 15 Substitute.

= 475 Simplify.

The probability that you will choose a dotted tile and then a dragon tile is 4

75.

3. In Problem 3, what is the probability that you randomly choose a bird and then, after replacing the first tile, a flower?

Two events are dependent events if the occurrence of one event affects the probability of the second event. For example, suppose in Problem 3 that you do not replace the first tile before choosing another. This changes the set of possible outcomes for your second selection.

Why are the events independent when you select with replacement?When you replace the tile, the conditions for the second selection are exactly the same as for the first selection.

Are the events independent?Yes. The outcome of rolling one number cube does not affect the outcome of rolling another number cube.


Problem 4

Got It?

Problem 5


Selecting Without Replacement

Games Suppose you choose a tile at random from the tiles shown in Problem 3. Without replacing the first tile, you select a second tile. What is the probability that you choose a dotted tile and then a dragon tile?

Because you do not replace the first tile, the events are dependent.


P (dragon after dotted) = 314 3 of the 14 remaining tiles are dragons.

P (dotted then dragon) = P (dotted) # P (dragon after dotted)

= 415

# 314 = 2

35 Substitute and then simplify.

The probability that you will choose a dotted tile and then a dragon tile is 235.

4. In Problem 4, what is the probability that you will randomly choose a flower and then, without replacing the first tile, a bird?

Finding the Probability of a Compound Event

Essay Contest One freshman, 2 sophomores, 4 juniors, and 5 seniors receive top scores in a school essay contest. To choose which 2 students will read their essays at the town fair, 2 names are chosen at random from a hat. What is the probability that a junior and then a senior are chosen?

The first outcome affects the probability of the second. So the events are dependent.

P (junior) = 412 = 1

3 4 of the 12 students are juniors.

P (senior after junior) = 511 5 of the 11 remaining students are seniors.

P (junior then senior) = P (junior) # P (senior after junior)

= 13# 5


The probability that a junior and then a senior are chosen is 533.

hsm11a1se_1208_t08505

9876543210

�

9876543210

9876543210

9876543210

9876543210

9876543210

2 / 53

Determine whether the events are dependent or independent and use the formula that applies.

P (junior then senior)Grade levels of the 12 students

Key Concept Probability of Two Dependent Events

If A and B are dependent events, P (A then B) = P (A) # P (B after A).

How is P(dragon after dotted) different from P(dragon)?After selecting the first tile without replacement, there is one less tile to choose from for the second choice.

HSM15_A2Reg_SE_CC_13_TrKit.indd Page 47 8/12/13 11:39 PM epg /120/PE01457/TRANSITION_KITS/NA/ANCILLARY/2015/XXXXXXXXXX/Layout/Interior_Files/S ...CC-13 Probability of Compound Events 47

HSM15_A2Reg_SE_TrKit.indd Page 47 8/13/13 12:27 AM epg /120/PE01457/TRANSITION_KITS/NA/ANCILLARY/2015/XXXXXXXXXX/Layout/Interior_Files/S ...

CC-13 47

Got It?

q What is the probability that you will choose a flower and then, without replacing the first tile, another flower? [The probability is zero.]

Problem 5

q If the first name were replaced prior to the second name being chosen, what might occur? [One student would need to read his or her essay twice.]

q What is the probability of choosing both of the sophomores to read their essays? Explain. [ 2

12# 1

11 = 2132 = 1

66]

AnswersGot It? (continued)

2. 2225

3. 118

4. 1105

Take NoteAsk students to state several more examples and a nonexample of dependent events.

Problem 4Show students that the probability can be computed using the Multiplication Counting Principle. The number of ways to choose a dotted tile first and then a dragon is 4 # 3 = 12 ways. The number of ways to choose two tiles is 15 # 14 = 210. Therefore, the probability is 12

210 = 235.

Got It?


5. a. In Problem 5, what is the probability that a senior and then a junior are chosen?

b. Reasoning Is P(junior then senior) different from P (senior then junior)? Explain.

Lesson CheckDo you know HOW?Use the cards below.

1. You choose a card at random. What is each probability?

a. P(B or number) b. P(red or 5)

c. P(red or yellow) d. P(yellow or letter)

2. What is the probability of choosing a yellow card and then a D if the first card is not replaced before the second card is drawn?

3. What is the probability of choosing a yellow card and then a D if the first card is replaced before the second card is drawn?

Do you UNDERSTAND? 4. Vocabulary What is an example of a compound

event composed of two overlapping events when you spin a spinner with the integers from 1 through 8?

5. Reasoning Are an event and its complement mutually exclusive or overlapping? Use an example to explain.

6. Open-Ended What is a real-world example of two independent events?

7. Error Analysis Describe and correct the error below in calculating P(yellow or letter) from Exercise 1, part (d).

c. wilsonhsm11a1se_1208_a07737

B 1 5 D 10

P(yellow or letter) = P(yellow) or P(letter)

= 1+ = 35

25

hsm11a1se_1208_t08514.aiPractice and Problem-Solving Exercises

You spin the spinner at the right, which is divided into equal sections. Find each probability.

8. P(4 or 7) 9. P(even or red) 10. P(odd or 10)

11. P(3 or red) 12. P(red or less than 3) 13. P(odd or multiple of 3)

14. P(7 or blue) 15. P(red or more than 8) 16. P(greater than 6 or blue)

You roll a blue number cube and a green number cube. Find each probability.

17. P(blue even and green even) 18. P(blue and green both less than 6)

19. P(green less than 7 and blue 4) 20. P(blue 1 or 2 and green 1)

You choose a tile at random from a bag containing 2 A’s, 3 B’s, and 4 C’s. You replace the first tile in the bag and then choose again. Find each probability.

21. P (A and A) 22. P (A and B) 23. P (B and B) 24. P (C and C) 25. P (B and C)

PracticeA See Problem 1.

3 107

2

496

8

5

1

hsm11a1se_1208_t08516.ai

See Problem 2.

See Problem 3.





You pick a coin at random from the set shown at the right and then pick a second coin without replacing the first. Find each probability.

26. P(dime then nickel) 27. P(quarter then penny)

28. P(penny then dime) 29. P(penny then quarter)

30. P(penny then nickel) 31. P(dime then penny)

32. P(dime then dime) 33. P(quarter then quarter)

34. Cafeteria Each day, you, Terry, and 3 other friends randomly choose one of your 5 names from a hat to decide who throws away everyone’s lunch trash. What is the probability that you are chosen on Monday and Terry is chosen on Tuesday?

35. Free Samples Samples of a new drink are handed out at random from a cooler holding 5 citrus drinks, 3 apple drinks, and 3 raspberry drinks. What is the probability that an apple drink and then a citrus drink are handed out?

Are the two events dependent or independent? Explain.

36. Toss a penny. Then toss a nickel.

37. Pick a name from a hat. Without replacement, pick a different name.

38. Pick a ball from a basket of yellow and pink balls. Return the ball and pick again.

39. Writing Use your own words to explain the difference between independent and dependent events. Give an example of each.

40. Reasoning A bag holds 20 yellow mints and 80 other green or pink mints. You choose a mint at random, eat it, and choose another.

a. Find the number of pink mints if P (yellow then pink) = P (green then yellow). b. What is the least number of pink mints if

P (yellow then pink) 7 P (green then yellow)?

41. Think About a Plan An acre of land is chosen at random from each of the three states listed in the table at the right. What is the probability that all three acres will be farmland?

• Does the choice of an acre from one state affect the choice from the other states?

• How must you rewrite the percents to use a formula from this lesson?

42. Phone Poll A pollster conducts a survey by phone. The probability that a call does not result in a person taking this survey is 85%. What is the probability that the pollster makes 4 calls and none result in a person taking the survey?

43. Open-Ended Find the number of left-handed students and the number of right-handed students in your class. Suppose your teacher randomly selects one student to take attendance and then a different student to work on a problem on the board.

a. What is the probability that both students are left-handed? b. What is the probability that both students are right-handed? c. What is the probability that the first student is right-handed and the second

student is left-handed?

See Problem 4.

See Problem 5.

ApplyB

Alabama

Florida

Indiana

27%

27%

65%

Percent of StateThat Is Farmland



48 Common Core

AnswersGot It? (continued)

5. a. 533

b. No; the numerators and the denominators are the same, so the product is the same.

Lesson Check 1. a. 4

5

b. 35 c. 1 d. 45 2. 3

20

3. 325

4. Answers may vary. Sample: find the probability of spinning a number less than 5 that is even.

5. Mutually exclusive; answers may vary. Sample: The complement of being even on a number die is being odd, and even and odd are mutually exclusive.

6. Check students’ work. 7. Because a tile can be both yellow

and a letter, the formula should be P (yellow or letter) = P (yellow) + P (letter) - P (yellow and letter) =

35 + 2

5 - 15 = 4

5.

Practice and Problem-Solving Exercises 8. 1

5 9. 45 10. 3

5 11. 12

12. 710 13. 3

5 14. 35 15. 7

10

16. 35 17. 1

4 18. 2536 19. 1

6

20. 118 21. 4

81 22. 227 23. 1

9

24. 1681 25. 4

27

3 Lesson CheckDo you know HOW?•IfstudentshavedifficultywithExercise3,then

have them review Problem 3 to understand how to handle replacement.

Do you UNDERSTAND?•IfstudentshavedifficultywithExercise4,then

have them also provide an example of a compound event composed of two mutually exclusive events when you spin a spinner with the integers from 1 through 8.

Close

q How does finding the probability of selecting with replacement compare to finding the probability of selecting without replacement? [When you select with replacement, the total number of possible outcomes is the same for each event. When you select without replacement, the total number of possible outcomes decreases after each event.]

q In problem 5, what is the probability that no seniors or juniors are chosen? [ 1

22]

Got It?

Got It?


5. a. In Problem 5, what is the probability that a senior and then a junior are chosen?

b. Reasoning Is P(junior then senior) different from P (senior then junior)? Explain.

Lesson CheckDo you know HOW?Use the cards below.

1. You choose a card at random. What is each probability?

a. P(B or number) b. P(red or 5)

c. P(red or yellow) d. P(yellow or letter)

2. What is the probability of choosing a yellow card and then a D if the first card is not replaced before the second card is drawn?

3. What is the probability of choosing a yellow card and then a D if the first card is replaced before the second card is drawn?

Do you UNDERSTAND? 4. Vocabulary What is an example of a compound

event composed of two overlapping events when you spin a spinner with the integers from 1 through 8?

5. Reasoning Are an event and its complement mutually exclusive or overlapping? Use an example to explain.

6. Open-Ended What is a real-world example of two independent events?

7. Error Analysis Describe and correct the error below in calculating P(yellow or letter) from Exercise 1, part (d).

c. wilsonhsm11a1se_1208_a07737

B 1 5 D 10

P(yellow or letter) = P(yellow) or P(letter)

= 1+ = 35

25

hsm11a1se_1208_t08514.aiPractice and Problem-Solving Exercises

You spin the spinner at the right, which is divided into equal sections. Find each probability.

8. P(4 or 7) 9. P(even or red) 10. P(odd or 10)

11. P(3 or red) 12. P(red or less than 3) 13. P(odd or multiple of 3)

14. P(7 or blue) 15. P(red or more than 8) 16. P(greater than 6 or blue)

You roll a blue number cube and a green number cube. Find each probability.

17. P(blue even and green even) 18. P(blue and green both less than 6)

19. P(green less than 7 and blue 4) 20. P(blue 1 or 2 and green 1)

You choose a tile at random from a bag containing 2 A’s, 3 B’s, and 4 C’s. You replace the first tile in the bag and then choose again. Find each probability.

21. P (A and A) 22. P (A and B) 23. P (B and B) 24. P (C and C) 25. P (B and C)

PracticeA See Problem 1.

3 107

2

496

8

5

1

hsm11a1se_1208_t08516.ai

See Problem 2.

See Problem 3.





You pick a coin at random from the set shown at the right and then pick a second coin without replacing the first. Find each probability.

26. P(dime then nickel) 27. P(quarter then penny)

28. P(penny then dime) 29. P(penny then quarter)

30. P(penny then nickel) 31. P(dime then penny)

32. P(dime then dime) 33. P(quarter then quarter)

34. Cafeteria Each day, you, Terry, and 3 other friends randomly choose one of your 5 names from a hat to decide who throws away everyone’s lunch trash. What is the probability that you are chosen on Monday and Terry is chosen on Tuesday?

35. Free Samples Samples of a new drink are handed out at random from a cooler holding 5 citrus drinks, 3 apple drinks, and 3 raspberry drinks. What is the probability that an apple drink and then a citrus drink are handed out?

Are the two events dependent or independent? Explain.

36. Toss a penny. Then toss a nickel.

37. Pick a name from a hat. Without replacement, pick a different name.

38. Pick a ball from a basket of yellow and pink balls. Return the ball and pick again.

39. Writing Use your own words to explain the difference between independent and dependent events. Give an example of each.

40. Reasoning A bag holds 20 yellow mints and 80 other green or pink mints. You choose a mint at random, eat it, and choose another.

a. Find the number of pink mints if P (yellow then pink) = P (green then yellow). b. What is the least number of pink mints if

P (yellow then pink) 7 P (green then yellow)?

41. Think About a Plan An acre of land is chosen at random from each of the three states listed in the table at the right. What is the probability that all three acres will be farmland?

• Does the choice of an acre from one state affect the choice from the other states?

• How must you rewrite the percents to use a formula from this lesson?

42. Phone Poll A pollster conducts a survey by phone. The probability that a call does not result in a person taking this survey is 85%. What is the probability that the pollster makes 4 calls and none result in a person taking the survey?

43. Open-Ended Find the number of left-handed students and the number of right-handed students in your class. Suppose your teacher randomly selects one student to take attendance and then a different student to work on a problem on the board.

a. What is the probability that both students are left-handed? b. What is the probability that both students are right-handed? c. What is the probability that the first student is right-handed and the second

student is left-handed?

See Problem 4.

See Problem 5.

ApplyB

Alabama

Florida

Indiana

27%

27%

65%

Percent of StateThat Is Farmland

HSM15_A2Reg_SE_CC_13_TrKit.indd Page 49 02/08/13 8:38 PM gg-018 /120/PE01457/TRANSITION_KITS/NA/ANCILLARY/2015/XXXXXXXXXX/Layout/Interior_Files/A ...CC-13 Probability of Compound Events 49


CC-13 49

26. 18 27. 1

36 28. 112 29. 1

36

30. 112 31. 1

12 32. 112 33. 0

34. 125 35. 3

22 36. Independent; the outcome of the

first event does not affect the second event.

37. Dependent;theoutcomeofthefirstevent affects the outcome of the second.

38. Independent; the outcome of the first event does not affect the second event.

39. For independent events, the outcome of the first event does not affect the outcome of the second event, while for dependent events, the outcome is affected. An example of two independent events is the rolling of two number cubes. An example of two dependent events is picking two cards from a deck without replacing the first one.

40. a. 40 pink mints b. 41 pink mints 41. about 4.7% 42. about 52.2% 43. a−c. Check students’ work.

4 PracticeaSSignMEnt gUidE

Basic: 8–35 all, 36–40 even, 41–42

Average: 9–35 odd, 36–43

Advanced: 9–35 odd, 36–45

Mathematical Practices are supported by exercises with red headings. Here are the Practices supported in this lesson:

MP 1: Make Sense of Problems Ex. 41MP 2: Reason Abstractly Ex. 5MP 2: Reason Quantitatively Ex. 40MP 3: Communicate Ex. 39MP 3: Critique the Reasoning of Others Ex. 7MP 4: Model with Mathematics Ex. 6, 43

Applications exercises have blue headings. Exercise 42 supports MP 4: Model.

HOMEWORK QUiCK CHECKTo check students’ understanding of key skills and concepts, go over Exercises 13, 27, 40, 41, and 42.


44. Suppose you roll a red number cube and a yellow number cube. a. What is P(red 1 and yellow 1)? b. What is P(red 2 and yellow 2)? c. What is the probability of rolling any matching pair of numbers? (Hint: Add the

probabilities of each of the six matches.)

45. A two-digit number is formed by randomly selecting from the digits 1, 2, 3, and 5 without replacement.

a. How many different two-digit numbers can be formed? b. What is the probability that a two-digit number contains a 2 or a 5? c. What is the probability that a two-digit number is prime?

ChallengeC



50 Common Core

AnswersPractice and Problem-Solving Exercises (continued)44. a. 1

36

b. 136

c. 16

45. a. 12

b. 56 c. 1

3

Documents

Common Core State Standards 1 Interactive Learning CC-13 ...terra-dade.enschool.org/ourpages/auto/2015/9/3/57271909/Alg II CC … · 03/09/2015 · an expression involving probabilities