common core state standards cc-13 probability of …...sep 29, 2015 · problem 2 problem 3 got it?...

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

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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.

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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.

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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.

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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.


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.

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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

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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.

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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


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

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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


common core state standards cc-13 probability of …...sep 29, 2015 · problem 2 problem 3 got it?...

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