Download Alg II CC-13 TE Probability of Compound Events

CC-13
1 Interactive Learning
Solve It!
PURPOSE To determine the probability of a
compound event using simple probability
PROCESS Students may use simple probability by
determining the number of favorable outcomes
and comparing it to the number of possible
outcomes.
Common Core State Standards
Probability of
Compound Events
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
Objectives To find probabilities of mutually exclusive and overlapping events
To find probabilities of independent and dependent events
FACILITATE
q How many songs on the music device are rock songs?
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?
Start with a plan.
How many songs
are there?
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]
Artist
Absolute Value
Algebras
Arithmetics
FOILs
Pascal’s Triangle
Pi
MATHEMATICAL
PRACTICES
ANSWER See Solve It in Answers on next page.
CONNECT THE MATH Students explore a compound
Category Songs
Rock
10
Pop
12
Rock
6
Pop
5
Country
12
Rock
11
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.
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.
Lesson
Vocabulary
•compound event
•mutually
exclusive events
•overlapping
events
•independent
events
•dependent
events
2 Guided Instruction
Take Note
Use Venn diagrams and the data provided in the
Solve It to illustrate the concepts of mutually
exclusive and overlapping events.
How man
are there
There are
2: 2, 4, 6, 8
16, 18, an
4 multiple
15, and 20
multiples o
and 20.
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).
Key Concept
Probability of A or B
Probability of Mutually Exclusive Events
If A and B are mutually exclusive events, P (A or B) = P(A) + P(B).
Probability of Overlapping Events
If A and B are overlapping events, P (A or B) = P (A) + P (B) - P (A and B).
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Chapter 12
Data Analysis and Probability
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CommonPage
Core
CC-13 Preparing to Teach
BIG idea Probability
Essential Understandings
•The probability of a compound event can
sometimes be found from expressions of the
probabilities of simpler events.
•Different methods must be used for finding the
probability of two dependent events compared
to finding the probability of two independent
events.
Math Background
A 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
44 Common Core
events are said to be independent if the
probability
of one event occurring is not
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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 Practice
Attend to precision. Students will
make explicit use of the terms “mutually
exclusive events” and “overlapping
events” and will determine when to
apply each.
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Problem 1 Mutually Exclusive and Overlapping Events
Problem 1
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?
q Is it possible for the spinner to land on both 2 and
Because the spinner cannot land on both 2 and 5, the events are mutually exclusive.
5 during the same spin? Explain. [No; there is only
P (2 or 5) = P (2) + P (5)
1
1
= 20
+ 20
2
1
= 20 = 10
one number on each section.]
q If you used the formula for overlapping events to
Substitute.
determine the probability in 1A, would you still
arrive at the same answer? [Yes; the probability
Simplify.
of landing on both 2 and 5 is zero, so you would
get the same answer.]
1
The probability that you spin a 2 or a 5 is 10
.
B What is the probability that you spin a number that is a multiple of 2 or 5?
q Is it possible to land on both a multiple of 2 and a
Since a number can be a multiple of 2 and a multiple of 5, such as 10, the events
are overlapping.
multiple of 5? Explain. [Yes; the numbers 10 and
20 are multiples of both 2 and 5.]
P (multiple of 2 or multiple of 5)
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.
q If you used the formula for mutually exclusive
= P (multiple of 2) + P (multiple of 5) - P (multiple of 2 and 5)
10
4
2
= 20 + 20
- 20
3
= 12
20 = 5
events to determine the probability in 1B, would
you still arrive at the same answer? [No, because
Substitute.
you would count some of the sections twice as
favorable outcomes.]
Simplify.
The probability that you spin a number that is a multiple of 2 or a multiple of 5 is
3
5.
Got It? 1. Suppose you roll a standard number cube.
Got It?
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?
q Which formula should you use to compute the
probability in 1a? 1b? [formula for overlapping
events; formula for mutually exclusive events]
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.
1st Choice
Red
2nd Choice
Red
Black
Black
Take Note
Ask students to state several more examples and a
nonexample of independent events.
Red
Black
Two events are independent events if the occurrence of one event does not affect the
probability of the second event.
hsm11a1se_1208_t08495.ai
Key Concept Probability of Two Independent Events
If A and B are independent events, P (A and B) = P(A)
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# P(B).
Probability of Compound Events
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Answers
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Solve It!
2
7;
explanations may vary.
Got It?
1.a. 56
b.23
CC-13 45
Problem 2 Finding the Probability of Independent Events
Problem 2
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?
The probability can be calculated by interpreting
this event as a simple event.
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
outcomes.]
# 6 = 36 possible
q What is the total number of favorable outcomes
1
P (red 3) = 6
3
Are the events
independent?
Yes. The outcome of
rolling one number
cube does not affect
the outcome of rolling
another number cube.
Three of the six numbers are even.
P (red 3 and blue even) = P (red 3)
= 16
The probability is
# P (blue even)
# 12 = 121
Substitute and then simplify.
1
12.
that you roll a 5 on the red cube and a 1 or 2 on the blue cube?
# 3 = 3 possible
Problem 3 Selecting With Replacement
Got It?
Ask students to describe an event involving the
number cubes that has a probability of 12 .
Problem 3
Show 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
4
225 = 75 .
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.
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?
4
P (dotted) = 15
4 of the 15 tiles are dotted.
3
P (dragon) = 15 = 15
3 of the 15 tiles are dragons.
P (dotted and dragon) = P (dotted)
4
= 15
4
= 75
# 15
# P (dragon)
Substitute.
Simplify.
The probability that you will choose a dotted tile and then a
4
.
dragon tile is 75
Got It? 3. In Problem 3, what is the probability that you randomly choose a bird and
Got It?
then, after replacing the first tile, a flower?
q What is the probability of choosing a bird tile? a
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.
2 1
; 15 ]
flower tile? [ 15
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Chapter 12
Data Analysis and Probability
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Additional Problems
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3.A bag contains 4 red chips,
5.Justin has 8 rock songs, 3 hip
3 green chips, 6 blue chips, and
hop songs, 5 classical music
5 black chips. Andrew selects a
songs, and 4 country songs in
chip at random. He replaces the
a playlist on his mp3 player.
chip and then selects another
Suppose he plays songs at
one at random. What is the
random from the playlist. If the
probability that he selects a red
mp3 player will not play the
chip, then a black chip?
same song twice in a row, what
5
is the probability that he will
Answer 81
hear a rock song followed by a
1
1
Answer a. 2 b. 3
4.Refer to the information
country song?
given in Additional Problem 3.
2.Suppose you roll a number cube
8
Answer 95
Suppose Andrew selects a chip
and flip a coin. What is the
at random, does not replace
probability of rolling a number
it, then selects another chip at
greater than 2 and flipping
random. What is the probability
heads?
that he selects a blue chip, then
1
Answer 3
a green chip?
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?
1
Answer 17
46 How is P(
dotted) d
from P(d
After selec
tile withou
there is on
to choose
second ch
Because you replace the first tile, the events are independent.
#
#
1
P (blue even) = 6 = 2
Got It? 2. You roll a red number cube and a blue number cube. What is the probability
possible when the two number cubes are rolled
simultaneously? [Using the Multiplication
Counting Principle, there are 1
favorable outcomes.]
Only one of the six numbers is a 3.
Common Core
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Take Note
Key Concept Probability of Two Dependent Events
If A and B are dependent events, P (A then B) = P (A)
# P (B after A).
Ask students to state several more examples and a
nonexample of dependent events.
Problem 4 Selecting Without Replacement
Problem 4
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.
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.
4
P (dotted) = 15
2 / 3 5
4 of the 15 tiles are dotted.
3
P (dragon after dotted) = 14
3 of the 14 remaining tiles are dragons.
# P (dragon after dotted)
# 143 = 352 Substitute and then simplify.
P (dotted then dragon) = P (dotted)
4
= 15
Show 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
2
210 = 35 .
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
#
Got It?
2
The probability that you will choose a dotted tile and then a dragon tile is 35
.
Got It? 4. In Problem 4, what is the probability that you will randomly choose a flower
hsm11a1se_1208_t08505
and then, without replacing the first tile, a bird?
Problem 5 Finding the Probability of a Compound Event
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?
2
1
2
1
Explain. [ 12
11 = 132 = 66 ]
#
The first outcome affects the probability of the second. So the events are dependent.
4
1
P (junior) = 12
=3
5
P (senior after junior) = 11
flower and then, without replacing the first tile,
another flower? [The probability is zero.]
q If the first name were replaced prior to the second
Determine whether the events are dependent or
independent and use the formula that applies.
P (junior then senior)
q What is the probability that you will choose a
Problem 5
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?
Grade levels of the
12 students
#
4 of the 12 students are juniors.
5 of the 11 remaining students are seniors.
P (junior then senior) = P (junior) # P (senior after junior)
5
5
= 13 # 11 = 33 Substitute and then simplify.
5
The probability that a junior and then a senior are chosen is 33
.
Lesson 12-8
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Got It? (continued)
2
2.225
1
3.18
1
4.105
CC-13 47
Got It? 5. a. In Problem 5, what is the probability that a senior and then a junior are
Got It?
chosen?
b. Reasoning Is P(junior then senior) different from
P (senior then junior)? Explain.
q In problem 5, what is the probability that no seniors
1
]
or juniors are chosen? [ 22
Lesson Check
3 Lesson Check
Do you know HOW?
B
•If students have difficulty with Exercise 3, then
have them review Problem 3 to understand how
to handle replacement.
1
5
D 10
5. Reasoning Are an event and its complement mutually
exclusive or overlapping? Use an example to explain.
1. You choose a card at random. What is each
probability?
Do you UNDERSTAND?
•If students have difficulty with Exercise 4, 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.
PRACTICES
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?
Use the cards below.
Do you know HOW?
MATHEMATICAL
Do you UNDERSTAND?
a. P(B or number)
b. P(red or 5)
c. P(red or yellow)
d. P(yellow or letter)
B
6. Open-Ended What is a real-world example of two
independent events?
App
7. Error Analysis Describe and correct the error below in
calculating P(yellow or letter) from Exercise 1, part (d).
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?
P(yellow or letter) = P(yellow) or P(letter)
=3 + 2
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?
5
=1
5
Close
q How does finding the probability of selecting with
replacement compare to finding the probability of
selecting without replacement? [When you select
MATHEMATICAL
Practice and Problem-Solving Exercises hsm11a1se_1208_t08514.ai
PRACTICES
A
Practice
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.]
8. P(4 or 7)
11. P(3 or red)
14. P(7 or blue)
9. P(even or red)
10. P(odd or 10)
12. P(red or less than 3)
15. P(red or more than 8)
13. P(odd or multiple of 3)
22. P (A and B)
23. P (B and B)
24. P (C and C)
9
4
See Problem 2.
hsm11a1se_1208_t08516.ai
See Problem 3.
25. P (B and C)
Data Analysis and Probability
Answers
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Got It? (continued)
5
5.a. 33
b.No; the numerators and the
denominators are the same, so the
product is the same.
Lesson Check
1.a. 45
b.35
c.1
d.45
3
3.25
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.
Common Core
6
20. P(blue 1 or 2 and green 1)
48
48 8
19. P(green less than 7 and blue 4)
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Common
Core
3
2.20
2
18. P(blue and green both less than 6)
21. P (A and A)
7
5
17. P(blue even and green even)
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.
Chapter 12
3 10
1
16. P(greater than 6 or blue)
You roll a blue number cube and a green number cube. Find each probability.
48
See Problem 1.
You spin the spinner at the right, which is divided into equal sections. Find
each probability.
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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) =
3
2
1
4
5 + 5 - 5 = 5.
Practice and Problem-Solving
Exercises
8. 15 9. 45 10. 35 11. 12
7
7
12. 10
13. 35 14. 35 15. 10
1
16. 35 17. 14 18. 25
36 19. 6
1
4
2
20. 18
21. 81
22. 27
23. 19
4
24. 16
81 25. 27
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)
4 Practice
28. P(penny then dime)
29. P(penny then quarter)
ASSIGNMENT GUIDE
30. P(penny then nickel)
31. P(dime then penny)
Basic: 8–35 all, 36–40 even, 41–42
32. P(dime then dime)
33. P(quarter then quarter)
Average: 9–35 odd, 36–43
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?
u
Advanced: 9–35 odd, 36–45
See Problem 5.
Mathematical Practices are supported by
exercises with red headings. Here are the Practices
supported in this lesson:
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?
lly
B
Apply
MP 1: Make Sense of Problems Ex. 41
MP 2: Reason Abstractly Ex. 5
MP 2: Reason Quantitatively Ex. 40
MP 3: Communicate Ex. 39
MP 3: Critique the Reasoning of Others Ex. 7
MP 4: Model with Mathematics Ex. 6, 43
Are the two events dependent or independent? Explain.
36. Toss a penny. Then toss a nickel.
in
).
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.
Applications exercises have blue headings.
Exercise 42 supports MP 4: Model.
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)?
em 1.
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?
HOMEWORK QUICK CHECK
To check students’ understanding of key skills and
concepts, go over Exercises 13, 27, 40, 41, and 42.
Percent of State
That Is Farmland
Alabama
27%
Florida
27%
Indiana
65%
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?
em 2.
208_t08516.ai
em 3.
See Problem 4.
43. Open-Ended Find the number of left-handed students and the number of righthanded 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?
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1
1
1
26. 18 27. 36
28. 12
29. 36
1
1
1
30. 12
31. 12
32. 12
33.0
3
1
34. 25 35. 22
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36. Independent; the outcome of the
first event does not affect the second
event.
37. Dependent; the outcome of the first
event 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.
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Answers
C
Challenge
Practice and Problem-Solving Exercises
(continued)
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?
1
44.a.36
1
b.36
c.16
45. a. 12
b.56
c.13
50
Chapter 12
Data Analysis and Probability
50
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Common
Core
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50 Common Core
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