Download Ch. 3 Probability 3.1 Events, Sample Spaces, and Probability

Ch. 3 Probability
3.1 Events, Sample Spaces, and Probability
1 List Sample Space and Assign Probabilities
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Which of the following assignments of probabilities to the sample points A, B, and C is valid if A, B, and C are
the only sample points in the experiment?
1
11
1
1
1
A) P(A) 0, P(B)
, P(C)
B) P(A)
, P(B)
, P(C)
12
12
7
2
8
C) P(A)
1
, P(B)
4
1
, P(C)
2
3
4
D) P(A)
1
, P(B)
4
1
, P(C)
4
1
4
2) If sample points A, B, C, and D are the only possible outcomes of an experiment, find the probability of D using
the table below.
Sample Point
Probability
8
A)
11
A
1/11
B
1/11
B)
1
11
C
1/11
D
.
C)
1
4
D)
3
11
3) A bag of candy was opened and the number of pieces was counted. The results are shown in the table below:
Color Number
Red
25
Brown
20
Green
20
Blue
15
Yellow
10
Orange
10
List the sample space for this problem.
A) {25, 20, 20, 15, 10, 10}
C) {Red, Brown, Green, Blue, Yellow, Orange}
B) {0.25, 0.20, 0.20, 0.15, 0.10, 0.10}
D) {Red}
4) A bag of candy was opened and the number of pieces was counted. The results are shown in the table below:
Color Number
Red
25
Brown
20
Green
20
Blue
15
Yellow
10
Orange
10
Find the probability that a randomly chosen piece of candy is not blue or red.
A) 0.40
B) 0.85
C) 0.15
D) 0.60
5) Fill in the blank. A(n) ______ is a process that leads to a single outcome that cannot be predicted with certainty.
A) experiment
B) event
C) sample point
D) sample space
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6) Fill in the blank. A(n) __________ is the most basic outcome of an experiment.
A) sample point
B) event
C) experiment
D) sample space
8) Fill in the blank. A(n) __________ is a collection of sample points.
A) event
B) Venn diagram
C) sample space
D) experiment
7) Fill in the blank. The __________ is the collection of all the sample points in an experiment.
A) sample space
B) Venn diagram
C) event
D) union
9) The outcome of an experiment is the number of resulting heads when a nickel and a dime are flipped
simultaneously. What is the sample space for this experiment?
A) {0, 1, 2}
B) {HH, HT, TT}
C) {HH, HT, TH, TT}
D) {nickel, dime}
10) A bag of colored candies contains 20 red, 25 yellow, and 35 orange candies. An experiment consists of
randomly choosing one candy from the bag and recording its color. What is the sample space for this
experiment?
A) {red, yellow, orange}
B) {20, 25, 35}
C) {1/4, 5/16, 7/16}
D) {80}
11) An experiment consists of rolling two dice and summing the resulting values. Which of the following is not a
sample point for this experiment?
A) 1
B) 2
C) 6
D) 7
12) Which number could be the probability of an event that occurs about as often as it does not occur?
A) .51
B) .51
C) 0
D) 1
13) Which number could be the probability of an event that rarely occurs?
A) .01
B) .01
C) .51
14) Which number could be the probability of an event that is almost certain to occur?
A) .99
B) 1.01
C) .51
D) .99
D) .01
15) Suppose that an experiment has five equally likely outcomes. What probability is assigned to each of the
sample points?
A) .2
B) .5
C) .05
D) 1
16) An experiment consists of randomly choosing a number between 1 and 10. Let E be the event that the number
chosen is even. List the sample points in E.
A) {2, 4, 6, 8, 10}
B) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C) {1, 3, 5, 7, 9}
D) {5}
Answer the question True or False.
17) A statistical experiment can be almost any act of observation as long as the outcome is uncertain.
A) True
B) False
18) The probability of a sample point is usually taken to be the relative frequency of the occurrence of the sample
point in a very long series of repetitions of the experiment.
A) True
B) False
19) In some experiments, we assign subjective probabilities, which can be interpreted as our degree of belief in the
outcome.
A) True
B) False
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20) In any experiment with exactly four sample points in the sample space, the probability of each sample point is
.25.
A) True
B) False
21) An event may contain sample points that are not in the original sample space of the experiment. For example,
the experiment of rolling two dice has the following sample space:
{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
However, the event of rolling a sum of at least 11 on the two dice is {11, 12}.
A) True
B) False
22) The probability of an event can be calculated by finding the sum of the probabilities of the individual sample
points in the event and dividing by the number of sample points in the event.
A) True
B) False
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
23) A package of self sticking notepads contains 6 yellow, 6 blue, 6 green, and 6 pink notepads. An experiment
consists of randomly selecting one of the notepads and recording its color. Find the sample space for the
experiment.
24) An experiment consists of randomly choosing a number between 1 and 10. Let A be the event that the number
chosen is less than or equal to 7. List the sample points in A.
25) An economy pack of highlighters contains 12 yellow, 6 blue, 4 green, and 3 orange highlighters. An experiment
consists of randomly selecting one of the highlighters. Find the probability that a blue highlighter is chosen.
26) Suppose that an experiment has eight equally likely outcomes. What probability is assigned to each of the
sample points?
2 Use Venn Diagram to Find Probability
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
1) The accompanying Venn diagram describes the sample space of a particular experiment and events A and B.
Suppose the sample points are equally likely. Find P(A) and P(B).
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2) The accompanying Venn diagram describes the sample space of a particular experiment and events A and B.
1
1
Suppose P(1) P(2) P(3) P(4)
and P(5) P(6) P(7) P(8) P(9) P(10)
. Find P(A) and P(B).
16
8
3 Find Probability Given Sample Space
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Probabilities of different types of vehicle to vehicle accidents are shown below:
Accident
Probability
Car to Car
0.66
Car to Truck
0.18
Truck to Truck 0.16
Find the probability that an accident involves a car.
A) 0.84
B) 0.66
C) 0.18
D) 0.16
C) 0.14
D) 0.34
2) A hospital reports that two patients have been admitted who have contracted Crohn's disease. Suppose our
experiment consists of observing whether each patient survives or dies as a result of the disease. The simple
events and probabilities of their occurrences are shown in the table (where S in the first position means that
patient 1 survives, D in the first position means that patient 1 dies, etc.).
Simple Events
SS
SD
DS
DD
Probabilities
0.52
0.20
0.14
0.14
Find the probability that both patients survive.
A) 0.52
B) 0.2704
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3) A hospital reports that two patients have been admitted who have contracted Crohn's disease. Suppose our
experiment consists of observing whether each patient survives or dies as a result of the disease. The simple
events and probabilities of their occurrences are shown in the table (where S in the first position means that
patient 1 survives, D in the first position means that patient 1 dies, etc.).
Simple Events
SS
SD
DS
DD
Probabilities
0.59
0.11
0.10
0.20
Find the probability that at least one of the patients does not survive.
A) 0.41
B) 0.21
C) 0.20
D) 0.11
4) A bag of candy was opened and the number of pieces was counted. The results are shown in the table below:
Color Number
Red
25
Brown
20
Green
20
Blue
15
Yellow
10
Orange
10
Find the probability that a randomly selected piece is either yellow or orange in color.
A) 20
B) 0.20
C) 10
D) 0.10
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
5) A hospital reports that two patients have been admitted who have contracted Crohn's disease. Suppose our
experiment consists of observing whether each patient survives or dies as a result of the disease. The simple
events and probabilities of their occurrences are shown in the table (where S in the first position means that
patient 1 survives, D in the first position means that patient 1 dies, etc.).
Simple Events
SS
SD
DS
DD
Probabilities
0.58
0.10
0.20
0.12
Find the probability that neither patient survives.
6) In a sample of 750 of its online customers, a department store found that 420 were men. Use this information to
estimate the probability that a randomly selected online customer is a man.
7) At a small private college with 800 students, 240 students receive some form of government sponsored
financial aid. Find the probability that a randomly selected student receives some form of
government sponsored financial aid.
8) The manager of a warehouse club estimates that 7 out of 10 customers will donate a dollar to help a children's
hospital during an annual drive to benefit the hospital. Using the manager's estimate, what is the probability
that a randomly selected customer will donate a dollar?
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9) A college has 85 male and 75 female fulltime faculty members. Suppose one fulltime faculty member is selected
at random and the faculty member's gender is observed.
a. List the sample points for this experiment.
b. Assign probabilities to the sample points.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
10) At a community college with 500 students, 120 students are age 30 or older. Find the probability that a
randomly selected student is age 30 or older.
A) .24
B) .76
C) .12
D) .30
11) A clothing vendor estimates that 78 out of every 100 of its online customers do not live within 50 miles of one
of its physical stores. Using this estimate, what is the probability that a randomly selected online customer does
not live within 50 miles of a physical store?
A) .78
B) .22
C) .50
D) .28
12) A music store has 8 male and 12 female employees. Suppose one employee is selected at random and the
employee's gender is observed. List the sample points for this experiment, and assign probabilities to the
sample points.
A) {male, female}; P(male) .4 and P(female) .6
B) {male, female}; P(male) .8 and P(female)
C) {8, 12}; P(8) .5 and P(12) .6
D) {8, 12}; P(8) .8 and P(12) .12
.12
13) An experiment consists of randomly choosing a number between 1 and 10. Let E be the event that the number
chosen is even. Assuming that each of the numbers between 1 and 10 is equally likely to be chosen, find P(E).
A) .5
B) .1
C) .2
D) .8
14) The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Find the
probability that the number rolled on a single roll of this die is less than 4.
Outcome
Probability
A) .3
1
.1
2
.1
3
.1
B) .2
4
.2
5
.2
C) .5
6
.3
D) .7
15) The table displays the probabilities for each of the outcomes when three fair coins are tossed and the number of
heads is counted. Find the probability that the number of heads on a single toss of the three coins is at most 2.
Outcome
Probability
A) .875
0
.125
1
.375
2
.375
B) .125
3
.125
C) .500
D) .750
16) At a certain university, one out of every 20 students is enrolled in a statistics course. If one student at the
university is chosen at random, what is the probability that the student is enrolled in a statistics course?
1
1
1
1
A)
B)
C)
D)
20
19
21
2
17) Two chips are drawn at random and without replacement from a bag containing four blue chips and three red
chips. Find the probability of drawing two red chips.
1
1
9
6
A)
B)
C)
D)
7
12
49
7
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
18) Three fair coins are tossed and either heads (H) or tails (T) is observed for each coin.
a.
b.
c.
d.
e.
List the sample points for this experiment.
Assign probabilities to the sample points.
Find the probability of the event A {Three heads are observed}.
Find the probability of the event B {Exactly two heads are observed}.
Find the probability of the event C {At least two heads are observed}.
a.
b.
c.
d.
e.
List the sample points for this experiment.
Assign probabilities to the sample points.
Find the probability of the event A {Two blue chips are drawn}.
Find the probability of the event B {A blue chip and a red chip are drawn}.
Find the probability of the event C {Two red chips are drawn}.
19) Two chips are drawn at random and without replacement from a bag containing three blue chips and one red
chip.
20) In an exit poll, 45% of voters said that the main issue affecting their choices of candidates was the economy,
35% said national security, and the remaining 20% were not sure. Suppose we select one of the voters who
participated in the exit poll at random and ask for the main issue affecting his or her choices of candidates.
a. List the sample points for this experiment.
b. Assign reasonable probabilities to the sample points.
c. Find the probability that the main issue affecting his or her choices was either the economy or national
security.
21) The data below show the types of medals won by athletes representing the United States in the Winter
Olympics. Suppose that one medal is chosen at random and the type of medal noted.
gold
bronze
gold
gold
a.
b.
c.
gold
gold
silver
gold
silver
silver
silver
bronze
gold
silver
bronze
bronze
bronze
bronze
bronze
silver
silver
gold
List the sample points for this experiment.
Find the probability of each sample point.
What is the probability that the medal was not bronze?
silver
gold
silver
22) The table shows the number of each type of book found at an online auction site during a recent search.
Suppose that Juanita randomly chose one book to bid on and then noted its type.
Type of Book
Children's
Fiction
Nonfiction
Educational
a.
b.
c.
Number
51,033
141,114
253,074
67,252
List the sample points for this experiment.
Find the probability of each sample point.
What is the probability that the book was nonfiction or educational?
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23) The table shows the number of each car sold in the United States in June. Suppose the sales record for one of
these cars is randomly selected and the type of car is identified.
Type of Car
Sedan
Convertible
Wagon
SUV
Van
Hatchback
Total
a.
b.
c.
Number
7,204
9,089
20,418
13,691
15,837
15,350
81,589
List the sample points for this experiment.
Find the probability of each sample point.
What is the probability that the car was a Van or an SUV?
24) The data show the total number of medals (gold, silver, and bronze) won by each country winning at least one
gold medal in the Winter Olympics. Suppose that one of the countries represented is chosen at random and the
total numbers of medals won by that country is noted.
1
11
a.
b.
c.
2
14
3
14
3
19
4
22
9
23
9
24
11
25
11
29
List the sample points for this experiment.
Find the probability of each sample point.
What is the probability that the country won at least 20 total medals?
25) The following data represent the scores of 50 students on a statistics exam. Suppose that one of the 50 students
is chosen at random and that student's score is noted.
39
71
79
85
90
a.
b.
c.
51
71
79
86
90
59
73
79
86
91
63
74
80
88
91
66
76
80
88
92
68
76
82
88
95
68
76
83
88
96
69
77
83
89
97
70
78
83
89
97
71
79
85
89
98
What is the probability that the student’s score is 88?
What is the probability that the student’s score is less than 60?
What is the probability that the student’s score is between 70 and 79, inclusive?
26) Three companies (A, B, and C) are to be ranked first, second, and third in a list of companies with the highest
customer satisfaction.
a. List all the possible sets of rankings for these top three companies.
b. Assuming that all sets of rankings are equally likely, what is the probability that Company A will be
ranked first, Company B second, and Company C third?
c. Assuming that all sets of rankings are equally likely, what is the probability that Company B will be ranked
first?
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4 Use Combination Rule
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Compute.
1)
8
2
2)
10
2
3)
7
7
4)
8
0
5)
9
8
A) 28
B) 56
C) 4
D) 720
A) 45
B) 90
C) 19
D) 8
A) 1
B) 6
C) 7
D) 720
A) 1
B) 7
C) 8
D) 5040
A) 9
B) 8
C) 362,880
D) 1
Compute the number of ways you can select n elements from N elements.
6) n 3, N 8
A) 56
B) 336
C) 3
7) n
4, N 10
A) 210
B) 5040
Solve the problem.
8) Which quantity is represented on the screen below?
C) 34
D) 120
D) 6
A) The number of ways two coins can be chosen from six coins
B) The number of sample points when a die is rolled and a coin is flipped
C) The number of ways two dice can be rolled
D) The number of sample points when a coin is flipped six times
9) Which expression is equal to
A)
N!
n!(N n)!
N
?
n
B)
N!
(N n)!
C)
N!
n!
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D)
N!
N!(N n)!
10) Evaluate
8
.
2
11) Evaluate
6
.
0
12) Evaluate
7
.
7
A) 28
A) 1
A) 1
B) 16
C) 56
D) 4
B) 6
C) 0
D) undefined
B) 7
C) 14
D) 49
13) Compute the number of ways you can select 3 elements from 7 elements.
A) 35
B) 21
C) 10
D) 343
14) There are 10 movies that Greg would like to rent but the store only allows him to have 4 movies at one time. In
how many ways can Greg choose 4 of the 10 movies?
A) 210
B) 40
C) 10,000
D) 5040
15) Kim submitted a list of 12 movies to an online movie rental company. The company will choose 3 of the movies
and ship them to her. If all movies are equally likely to be chosen, what is the probability that Kim will receive
the three movies that she most wants to watch? Express the probability as a fraction.
1
1
1
1
A)
B)
C)
D)
220
4
1320
1728
Answer the question True or False.
16) The quantity 0! is defined to be equal to 0.
A) True
B) False
17) The combinations rule applies to situations in which the experiment calls for selecting n elements from a total
of N elements, without replacing each element before the next is selected.
A) True
B) False
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
18) Compute
10
.
6
19) Compute
5
.
1
20) Compute the number of ways you can select n elements from N elements for n
6 and N
15.
21) In how many ways can a manager choose 3 of his 8 employees to work overtime helping with inventory?
22) The manager of an advertising department has asked her creative team to propose six new ideas for an
advertising campaign for a major client. She will choose three of the six proposals to present to the client. (We
will refer to the six proposals as A, B, C, D, E, and F.)
a. In how many ways can the manager select the three of the six proposals? List the possibilities.
b. It is unlikely that the manager will randomly select three of the six proposals, but if she does what is the
probability that she selects proposals A, D, and E?
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3.2 Unions and Intersections
1 Find Union or Intersection of Events
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows.
A: {The number is even}
B: {The number is less than 7}
Identify the sample points in the event A B.
A) {1, 2, 3, 4, 5, 6, 8, 10}
C) {1, 2, 3, 4, 5, 6, 7, 8, 10}
B) {2, 4, 6}
D) {1, 2, 3, 4, 5, 6, 7, 9}
2) A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows.
A: {The number is even}
B: {The number is less than 7}
Identify the sample points in the event A
A) {2, 4, 6}
C) {1, 2, 3, 4, 5, 6, 7, 8, 10}
B.
B) {1, 2, 3, 4, 5, 6, 8, 10}
D) {1, 2, 3, 4, 5, 6, 7, 9}
3) A pair of fair dice is tossed. Events A and B are defined as follows.
A: {The sum of the numbers on the dice is 3}
B: {At least one of the dice shows a 2}
Identify the sample points in the event A B.
A) {(1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)}
B) {(1, 2), (2, 1)}
C) {(1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)}
D) {(2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)}
4) A pair of fair dice is tossed. Events A and B are defined as follows.
A: {The sum of the numbers on the dice is 3}
B: {At least one of the dice shows a 2}
Identify the sample points in the event A B.
A) {(1, 2), (2, 1)}
B) {(1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)}
C) {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)}
D) {(2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)}
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5) A pair of fair dice is tossed. Events A and B are defined as follows.
A: {The sum of the numbers on the dice is 4}
B: {The sum of the numbers on the dice is 11}
Identify the sample points in the event A B.
A) {(1, 3), (2, 2), (3, 1), (5, 6), (6, 5)}
C) {(1, 4), (2, 3), (3, 2), (4, 1), (5, 6), (6, 5)}
B) {(1, 4), (2, 2), (4, 1), (5, 6), (6, 5)}
D) There are no sample points in the event A B.
6) A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows.
A: {The number is even}
B: {The number is less than 7}
Which expression represents the event that the number is even or less than 7 or both?
A) A B
B) A B
C) Ac
D) Bc
7) A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows.
A: {The number is even}
B: {The number is less than 7}
Which expression represents the event that the number is both even and less than 7?
A) A B
B) A B
C) Ac
D) Bc
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
8) A company evaluates its potential new employees using three criteria.
A: The applicant has a minimum college GPA of 3.0.
B: The applicant has relevant work experience.
C: The applicant has a sufficient score on an aptitude test.
a. Write the event that an applicant meets all three criteria as a union or intersection of A, B, and C.
b. Write the event that an applicant meets at least one of the three criteria as a union or intersection of A, B,
and C.
9) A consumer advocacy group rates the quality of a cellular service provider using three criteria.
A: Service is available at least 99% of the time.
B: Reception is clear at least 95% of the time.
C: Fewer than 5% of its customers have complaints about the quality of service.
a.
b.
Describe the event represented by A B C.
Describe the event represented by A B C.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
10) Fill in the blank. The __________ of two events A and B is the event that either A or B or both occur.
A) union
B) intersection
C) complement
D) Venn diagram
11) Fill in the blank. The __________ of two events A and B is the event that both A and B occur.
A) intersection
B) union
C) complement
D) Venn diagram
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Answer the question True or False.
12) Unions and intersections of events are examples of compound events.
A) True
B) False
13) Unions and intersections cannot be defined for more than two sets, so that A B C and A
meaningless.
A) True
B) False
B
C are
14) A pair of fair dice is tossed. Events A and B are defined as follows.
A: {The sum of the numbers on the dice is 3}
B: {At least one of the dice shows a 2}
True or False: A B
A) True
B.
B) False
15) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red
chips. Events A and B are defined as follows.
A: {Both chips are red}
B: {At least one of the chips is blue}
True or False: A
A) True
B
B.
B) False
2 Find Probability Using Unions and Intersections
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) The overnight shipping business has skyrocketed in the last ten years. The single greatest predictor of a
company's success is customer service. A study was conducted to determine the customer satisfaction levels for
one overnight shipping business. In addition to the customer's satisfaction level, the customers were asked how
often they used overnight shipping. The results are shown in the following table:
Frequency of Use
2 per month
2 5 per month
5 per month
TOTAL
High
250
140
70
460
Satisfaction level
Medium
140
55
25
220
Low
10
5
5
20
TOTAL
400
200
100
700
Suppose that one customer who participated in the study is chosen at random. What is the probability that the
customer had a medium level of satisfaction and used the company more than five times per month?
1
16
59
81
A)
B)
C)
D)
28
35
140
140
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2) Each manager of a Fortune 500 company was rated as being either a good, fair, or poor manager by his/her
boss. The manager's educational background was also noted. The data appear below:
Educational Background
Manager
Rating
H. S. Degree Some College College Degree Master's or Ph.D. Total
Good
5
6
21
7
39
Fair
4
12
48
23
87
Poor
7
2
3
22
34
Total
16
20
72
52
160
What is the probability that a randomly chosen manager has earned at least one college degree?
31
9
13
9
A)
B)
C)
D)
40
20
40
40
3) Each manager of a corporation was rated as being either a good, fair, or poor manager by his/her boss. The
manager's educational background was also noted. The data appear below:
Educational Background
Manager
Rating H. S. Degree Some College College Degree Master's or Ph.D. Totals
Good
2
8
24
5
39
Fair
3
15
49
20
87
Poor
1
9
6
18
34
Totals
6
32
79
43
160
If we randomly selected one manager from this company, find the probability that he or she has an advanced
(Master's or Ph.D.) degree and is a good manager.
1
29
41
31
B)
C)
D)
A)
40
80
32
32
4) Four hundred accidents that occurred on a Saturday night were analyzed. The number of vehicles involved
and whether alcohol played a role in the accident were recorded. The results are shown below:
Number of Vehicles Involved
Did Alcohol Play a Role?
1
2
3 or more Totals
Yes
59
98
13
170
No
20
171
39
230
Totals
79
269
52
400
Suppose that one of the 400 accidents is chosen at random. What is the probability that the accident involved
more than a single vehicle?
321
13
79
13
A)
B)
C)
D)
400
100
400
400
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5) A fast food restaurant chain with 700 outlets in the United States has recorded the geographic location of its
restaurants in the accompanying table of percentages. One restaurant is to be chosen at random from the 700 to
test market a new chicken sandwich.
Region
NE
SE
SW
NW
10,000
5%
6%
3%
0%
Population of City 10,000 100,000
15%
10%
12%
5%
100,000
20%
4%
9%
11%
What is the probability that the restaurant is located in the northern portion of the United States?
A) 0.56
B) 0.40
C) 0.16
D) 0.44
6) A fast food restaurant chain with 700 outlets in the United States has recorded the geographic location of its
restaurants in the accompanying table of percentages. One restaurant is to be chosen at random from the 700 to
test market a new chicken sandwich.
Region
NE
SE
SW
NW
10,000
8%
6%
3%
0%
Population of City 10,000 100,000
15%
7%
12%
5%
100,000
20%
4%
9%
11%
What is the probability that the restaurant is located in a city with a population over 100,000 and in the
southern portion of the United States?
A) 0.13
B) 0.04
C) 0.09
D) 0.41
7) The table shows the political affiliations and types of jobs for workers in a particular state. Suppose a worker is
selected at random within the state and the worker's political affiliation and type of job are noted.
Political Affiliation
Republican
Democrat
Independent
White collar
20%
7%
11%
Type of job
Blue Collar
17%
19%
26%
Find the probability that the worker is a white collar worker affiliated with the Democratic Party.
A) 0.07
B) 0.38
C) 0.26
D) 0.57
8) The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Suppose
that the die is rolled once. Let A be the event that the number rolled is less than 4, and let B be the event that the
number rolled is odd. Find P(A B).
Outcome
Probability
A) .5
1
.1
2
.1
B) .2
3
.1
4
.2
5
.2
C) .3
6
.3
D) .7
9) The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Suppose
that the die is rolled once. Let A be the event that the number rolled is less than 4, and let B be the event that the
number rolled is odd. Find P(A B).
Outcome
Probability
A) .2
1
.1
2
.1
B) .5
3
.1
4
.2
5
.2
C) .3
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6
.3
D) .7
10) A sample of 350 students was selected and each was asked the make of their automobile (foreign or domestic)
and their year in college (freshman, sophomore, junior, or senior). The results are shown in the table below.
Find the probability that a randomly selected student is both a sophomore and drives a foreign automobile.
A) 65/205
B) 65/350
C) 65/110
D) 45/350
11) A sample of 350 students was selected and each was asked the make of their automobile (foreign or domestic)
and their year in college (freshman, sophomore, junior, or senior). The results are shown in the table below.
What is the probability of randomly selecting a student who is in the freshman class or drives a foreign
automobile?
A) 215/350
B) 230/350
C) 15/205
D) 15/350
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
12) Suppose that an experiment has five sample points, E1 , E2 , E3 , E4 , E5 , and that P(E1 ) .2, P(E2 ) .3,
P(E3 ) .1, P(E4 ) .1, and P(E5 ) .3. If the events A and B are defined as A {E1 , E2 , E3 } and B {E2 , E3 , E4 }
find P(A B).
13) Suppose that an experiment has five sample points, E1 , E2 , E3 , E4 , E5 , and that P(E1 ) .4, P(E2 ) .1,
P(E3 ) .1, P(E4 ) .2, and P(E5 ) .2. If the events A and B are defined as A {E1 , E2 , E5 } and B {E2 , E3 , E5 }
find P(A B).
14) A fast food restaurant chain with 700 outlets in the United States has recorded the geographic location of its
restaurants in the accompanying table of percentages. One restaurant is to be chosen at random from the 700 to
test market a chicken sandwich.
Region
NE
SE
SW
NW
10,000
7%
6%
3%
0%
Population of City 10,000 100,000
15%
4%
12%
5%
100,000
20%
4%
1%
23%
What is the probability that the restaurant is located in the western portion of the United States?
15) The table shows the political affiliations and types of jobs for workers in a particular state. Suppose a worker is
selected at random within the state and the worker's political affiliation and type of job are noted.
Political Affiliation
Republican
Democrat
Independent
White collar
9%
19%
14%
Type of job
Blue Collar
15%
12%
31%
What is the probability that the worker is a white collar Republican?
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16) A pair of fair dice is tossed. Events A and B are defined as follows.
A: {The sum of the numbers on the dice is 6}
B: {At least one of the numbers 3}
a.
b.
c.
d.
Identify the sample points in the event A B.
Identify the sample points in the event A B.
Find P(A B).
Find P(A B).
17) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red
chips. Events A and B are defined as follows.
A: {Both chips are red}
B: {At least one of the chips is blue}
a.
b.
Identify the sample points in the event A B.
Find P(A B).
18) The table shows the number of each Ford car sold in the United States in June. Suppose the sales record for one
of these cars is randomly selected and the type of car is identified.
Type of Car
Sedan
Convertible
Wagon
SUV
Van
Hatchback
Total
Number
7,204
9,089
20,418
13,691
15,837
15,350
81,589
Events A and B are defined as follows.
A: {Convertible, SUV, Van}
B: {Fewer than 10,000 of the type of car were sold in June}
a.
b.
c.
d.
Identify the sample points in the event A B.
Identify the sample points in the event A B.
Find P(A B).
Find P(A B).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
19) In the game of Parcheesi each player rolls a pair of dice on each turn. In order to begin the game, you must roll
a five on at least one die, or a total of five on both dice. Find the probability that a player begins the game on
the first roll.
15
11
5
1
A)
B)
C)
D)
36
36
18
6
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3.3 Complementary Events
1 Find Complement of Event
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Fill in the blank. The __________ of an event A is the event that A does not occur.
A) complement
B) intersection
C) union
D) Venn diagram
2) The following Venn diagram shows the six possible outcomes when rolling a fair die. Let A be the event of
rolling an even number and let B be the event of rolling a number greater than 1.
Which of the following expressions describes the event of rolling a 1?
A) Bc
B) Ac
C) B
D) A B
3) A state energy agency mailed questionnaires on energy conservation to 1,000 homeowners in the state capital.
Five hundred questionnaires were returned. Suppose an experiment consists of randomly selecting one of the
returned questionnaires. Consider the events:
A: {The home is constructed of brick}
B: {The home is more than 30 years old}
In terms of A and B, describe a home that is constructed of brick and is less than or equal to 30 years old.
A) A Bc
B) A B
C) A B
D) (A B)c
4) A state energy agency mailed questionnaires on energy conservation to 1,000 homeowners in the state capital.
Five hundred questionnaires were returned. Suppose an experiment consists of randomly selecting one of the
returned questionnaires. Consider the events:
A: {The home is constructed of brick}
B: {The home is more than 30 years old}
D: {The home is heated with oil}
Which of the following describes the event B Dc?
A) homes more than 30 years old or homes that are not heated with oil
B) homes more than 30 years old that are heated with oil
C) homes that are not older than 30 years old and heated with oil
D) homes more than 30 years old that are not heated with oil
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5) An insurance company looks at many factors when determining how much insurance will cost for a home.
Two of the factors are listed below:
A: {The home's roof is less than 10 years old}
B: {The home has a security system}
In the words of the problem, define the event Bc.
A) The home is less than 10 years old
C) The home has a security system
B) The home is not less than 10 years old
D) The home does not have a security system
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
6) A fair die is rolled one time. Let A be the event that an odd number is rolled. Describe the event Ac.
7) A fair die is rolled one time. Let B be the event {1, 2, 5}. List the sample points in the event Bc.
8) A company evaluates its potential new employees using three criteria.
A: The applicant has a minimum college GPA of 3.0.
B: The applicant has relevant work experience.
D: The applicant has a sufficient score on an aptitude test.
Describe an applicant represented by A
Bc
D.
9) A consumer advocacy group rates the quality of a cellular service provider using three criteria.
A: Service is available at least 99% of the time.
B: Reception is clear at least 95% of the time.
D: Fewer than 5% of its customers have complaints about the quality of service.
Describe a cellular service provider represented by Ac Bc Dc.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Answer the question True or False.
10) If an event A includes the entire sample space, then P(Ac)
A) True
0.
B) False
11) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red
chips. Events A and B are defined as follows.
A: {Both chips are red}
B: {At least one of the chips is blue}
True or False: A
A) True
Bc.
B) False
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2 Find Probability Using Complement
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) At a community college with 500 students, 120 students are age 30 or older. Find the probability that a
randomly selected student is less than 30 years old.
A) .76
B) .24
C) .12
D) .30
2) A clothing vendor estimates that 78 out of every 100 of its online customers do not live within 50 miles of one
of its physical stores. Using this estimate, what is that probability that a a randomly selected online customer
lives within 50 miles of a physical store?
A) .22
B) .78
C) .50
D) .28
3) The overnight shipping business has skyrocketed in the last ten years. The single greatest predictor of a
company's success is customer service. A study was conducted to determine the customer satisfaction levels for
one overnight shipping business. In addition to the customer's satisfaction level, the customers were asked how
often they used overnight shipping. The results are shown below in the following table:
Frequency of Use
2 per month
2 5 per month
5 per month
TOTAL
High
250
140
70
460
Satisfaction level
Medium
140
55
25
220
Low
10
5
5
20
TOTAL
400
200
100
700
Suppose that one customer who participated in the study is chosen at random. What is the probability that the
customer did not have a high level of satisfaction with the company?
12
23
4
3
A)
B)
C)
D)
35
35
7
7
4) The table shows the political affiliations and types of jobs for workers in a particular state. Suppose a worker is
selected at random within the state and the worker's political affiliation and type of job are noted.
Political Affiliation
Republican
Democrat
Independent
White collar
14%
19%
16%
Type of job
Blue Collar
10%
9%
32%
Find the probability the worker is not an Independent.
A) 0.52
B) 0.48
C) 0.33
D) 0.19
5) A local country club has a membership of 600 and operates facilities that include an 18 hole championship golf
course and 12 tennis courts. Before deciding whether to accept new members, the club president would like to
know how many members regularly use each facility. A survey of the membership indicates that 67% regularly
use the golf course, 47% regularly use the tennis courts, and 6% use neither of these facilities regularly. What is
the probability that a member regularly uses at least one of the golf or tennis facilities?
A) .94
B) .6
C) .47
D) .20
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
6) Suppose that for a certain experiment P(A)
.37. Find P(Ac).
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7) Suppose that for a certain experiment the probability of a particular event occurring is .21. Find the probability
that this event does not occur.
8) Suppose that an experiment has five sample points, E1 , E2 , E3 , E4 , E5 , and that P(E1 )
P(E3 )
.1, P(E4 )
.1, and P(E5 )
.3. If event A is defined as A
{E1 , E2 , E3 }, find P(Ac).
.2, P(E2 )
.3,
9) In a sample of 750 of its online customers, a department store found that 420 were men. Use this information to
estimate the probability that a randomly selected online customer is not a man.
10) At a small private college with 800 students, 240 students receive some form of government sponsored
financial aid. Find the probability that a randomly selected student does not receive some form of
government sponsored financial aid.
11) The manager of a warehouse club estimates that 7 out of 10 customers will donate a dollar to help a children's
hospital during an annual drive to benefit the hospital. Using the manager's estimate, what is the probability
that a randomly selected customer will not donate a dollar?
12) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red
chips. Event A is defined as follows.
A: {Both chips are red}
a.
b.
c.
Describe the event Ac.
Identify the sample points in the event Ac.
Find P(Ac).
13) The table shows the number of each Ford car sold in the United States in June. Suppose the sales record for one
of these cars is randomly selected and the type of car is identified.
Type of Car
Sedan
Convertible
Wagon
SUV
Van
Hatchback
Total
Number
7,204
9,089
20,418
13,691
15,837
15,350
81,589
Event A is defined as follows.
A: {Convertible, SUV, Van}
a.
b.
Identify the sample points in the event Ac.
Find P(Ac).
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14) A pair of fair dice is tossed. Events A and B are defined as follows.
A: {The two numbers rolled are different}
B: {At least one of the numbers is greater than 2}
b.
Identify the sample points in the event Ac.
Identify the sample points in the event Bc.
f.
Find P(Ac
a.
Identify the sample points in the event Ac Bc.
d. Identify the sample points in the event Ac Bc.
e. Find P(Ac Bc).
c.
Bc).
3.4 The Additive Rule and Mutually Exclusive Events
1 Determine if Events are Mutually Exclusive
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) A sample of 350 students was selected and each was asked the make of their automobile (foreign or domestic)
and their year in college (freshman, sophomore, junior, or senior). The results are shown in the table below.
Which of the following events listed would be considered mutually exclusive events?
A) The student is a freshman and the student drives a foreign automobile
B) The student is a junior and the student drives a domestic automobile
C) The student is a junior and the student is a freshman
D) The student is a senior and the student drives a domestic automobile.
2) If P(A B) 1 and P(A B) 0, then which statement is true?
A) A and B are complementary events.
B) A and B are supplementary events.
C) A and B are both empty events.
D) A and B are reciprocal events.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
3) A pair of fair dice is tossed. Events A and B are defined as follows.
A: {The two numbers rolled are different}
B: {At least one of the numbers is greater than 2}
Are the events A and B mutually exclusive? Explain.
4) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red
chips. Events A and B are defined as follows.
A: {Both chips are red}
B: {At least one of the chips is blue}
Are the events A and B mutually exclusive? Explain.
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5) A number between 1 and 10, inclusive, is randomly chosen. Events A, B, C, and D are defined as follows.
A: {The number is even}
B: {The number is less than 7}
C: {The number is odd}
D: {The number is greater than 5}
Identify one pair of mutually exclusive events.
6) Three fair coins are tossed and either heads or tails is observed for each coin. Events A and B are defined as
follows.
A: {Three heads are observed}.
B: {Exactly two heads are observed}.
Is P(A
B) equal to the sum of P(A) and P(B)? Explain.
7) The table shows the number of each Ford car sold in the United States in June. Suppose the sales record for one
of these cars is randomly selected and the type of car is identified.
Type of Car
Sedan
Convertible
Wagon
SUV
Van
Hatchback
Total
Number
7,204
9,089
20,418
13,691
15,837
15,350
81,589
Events A and B are defined as follows.
A: {Convertible, SUV, Van}
B: {Fewer than 10,000 of the type of car were sold in June}
Is P(A
B) equal to the sum of P(A) and P(B)? Explain.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Answer the question True or False.
8) If two events, A and B, are mutually exclusive, then P(A and B) P(A)
A) True
B) False
9) An event and its complement are mutually exclusive.
A) True
10) If A and B are mutually exclusive events, then P(A B)
A) True
0.
P(B).
B) False
B) False
11) If events A and B are not mutually exclusive, then it is possible that P(A)
A) True
B) False
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P(B)
1.
2 Use Additive Rule to Find Probability
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Suppose that for a certain experiment P(A)
P(A B).
A) .62
B) .38
2) Suppose that for a certain experiment P(A)
A) .58
B) .72
.33 and P(B)
.29. If A and B are mutually exclusive events, find
.47 and P(B)
.25 and P(A
C) .86
C) .03
B)
D) .31
.14. Find P(A B).
D) .36
3) In a class of 40 students, 22 are women, 10 are earning an A, and 7 are women that are earning an A. If a
student is randomly selected from the class, find the probability that the student is a woman or earning an A.
A) .625
B) .8
C) .975
D) .25
4) In a class of 30 students, 18 are men, 6 are earning a B, and no men are earning a B. If a student is randomly
selected from the class, find the probability that the student is a man or earning a B.
A) .8
B) .4
C) .54
D) .24
5) In a box of 50 markers, 30 markers are either red or black and 20 are missing their caps. If 12 markers are either
red or black and are missing their caps, find the probability that a randomly selected marker is red or black or
is missing its cap.
A) .76
B) 1
C) .38
D) .24
6) In a box of 75 markers, 36 markers are either red or black and 15 are blue. Find the probability that a randomly
selected marker is red or black or blue.
A) .68
B) .51
C) .32
D) .24
7) Each manager of a corporation was rated as being either a good, fair, or poor manager by his/her boss. The
manager's educational background was also noted. The data appear below:
Educational Background
Manager
Rating H. S. Degree Some College College Degree Master's or Ph.D. Totals
Good
4
1
21
13
39
Fair
5
12
43
27
87
Poor
2
9
8
15
34
Totals
11
22
72
55
160
What is the probability that a randomly chosen manager is either a good managers or has an advanced degree?
81
13
47
147
A)
B)
C)
D)
160
160
80
160
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8) Four hundred accidents that occurred on a Saturday night were analyzed. The number of vehicles involved
and whether alcohol played a role in the accident were recorded. The results are shown below:
Number of Vehicles Involved
Did Alcohol Play a Role?
1
2
3 or more Totals
Yes
54
97
19
170
No
27
174
29
230
Totals
81
271
48
400
Suppose that one of the 400 accidents is chosen at random. What is the probability that the accident involved
alcohol or a single car?
197
17
81
27
A)
B)
C)
D)
400
40
400
200
9) A medium sized company characterized their employees based on the sex of the employee and their length of
service to the company. The results are summarized in the table below.
What proportion of the employees are female or have been employed for more than 10 years?
A) 85/130
B) 110/130
C) 25/130
D) 25/65
10) A medium sized company characterized their employees based on the sex of the employee and their length of
service to the company. The results are summarized in the table below.
What proportion of the employees are male or have been employed for less than 11 years?
A) 45/130
B) 165/130
C) 42/65
D) 120/130
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
11) A local country club has a membership of 600 and operates facilities that include an 18 hole championship golf
course and 12 tennis courts. Before deciding whether to accept new members, the club president would like to
know how many members regularly use each facility. A survey of the membership indicates that 68% regularly
use the golf course, 43% regularly use the tennis courts, and 14% use both of these facilities regularly. Find the
probability that a randomly selected member uses the golf or tennis facilities regularly.
12) Suppose that for a certain experiment P(A)
.37, P(B)
13) Suppose that for a certain experiment P(A)
1
and P(B)
3
P(A
B).
.69, and P(A
B)
.23. Find P(A
B).
1
, and events A and B are mutually exclusive. Find
4
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14) Suppose that for a certain experiment P(A)
A and B can not be mutually exclusive.
.8 and P(B)
.9. Use the Additive Rule to explain why the events
15) Based on past experience, Josh believes that the probability of catching a red snapper is .21 and the probability
of catching a grouper is .19. Is enough information available to find the probability of catching a red snapper or
a grouper? Explain. If possible, find the probability of catching a red snapper or a grouper.
16) Based on past experience, Josh believes that the probability of catching a red snapper is .21 and the probability
of catching a fish that weighs less than 5 pounds is .45. Is enough information available to find the probability
of catching a red snapper or a fish that weighs less than 5 pounds? Explain. If possible, find the probability of
catching a red snapper or a fish that weighs less than 5 pounds.
17) Suppose that 62% of the employees at a company are male and that 35% of the employees just received merit
raises. If 20% of the employees are male and received a merit raise, what is the probability that a randomly
chosen employee is male or received a merit raise?
18) Suppose that 80% of the employees of a company received cash or company stock as a bonus at the end of the
year. If 60% of the employees received a cash bonus and 30% received stock, what is the probability that a
randomly chosen employee received both cash and stock as a bonus?
3.5 Conditional Probability
1 Find Conditional Probability
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) A package of self sticking notepads contains 6 yellow, 6 blue, 6 green, and 6 pink notepads. An experiment
consists of randomly selecting one of the notepads and recording its color. Find the probability that a green
notepad is selected given that it is either blue or green.
1
1
1
1
A)
B)
C)
D)
2
3
4
12
2) A package of self sticking notepads contains 6 yellow, 6 blue, 6 green, and 6 pink notepads. An experiment
consists of randomly selecting one of the notepads and recording its color. Find the probability that a yellow or
pink notepad is selected given that it is either blue or green.
1
1
A) 0
B)
C)
D) 1
2
4
3) An economy pack of highlighters contains 12 yellow, 6 blue, 4 green, and 3 orange highlighters. An experiment
consists of randomly selecting one of the highlighters and recording its color. Find the probability that a blue or
yellow highlighter is selected given that a yellow highlighter is selected.
1
1
A) 1
B)
C)
D) 0
2
3
4) In a class of 40 students, 22 are women, 10 are earning an A, and 7 are women that are earning an A. If a
student is randomly selected from the class, find the probability that the student is a woman given that the
student is earning an A.
7
7
5
11
A)
B)
C)
D)
22
11
20
10
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5) In a class of 40 students, 22 are women, 10 are earning an A, and 7 are women that are earning an A. If a
student is randomly selected from the class, find the probability that the student is earning an A given that the
student is a woman.
7
1
5
7
A)
B)
C)
D)
22
4
11
40
6) In a class of 30 students, 18 are men, 6 are earning a B, and no men are earning a B. If a student is randomly
selected from the class, find the probability that the student is a man given that the student earning a B.
3
1
A) 0
B)
C)
D) 1
5
3
7) The overnight shipping business has skyrocketed in the last ten years. The single greatest predictor of a
company's success is customer service. A study was conducted to determine the customer satisfaction levels for
one overnight shipping business. In addition to the customer's satisfaction level, the customers were asked how
often they used overnight shipping. The results are shown below in the following table:
Frequency of Use
2 per month
2 5 per month
5 per month
TOTAL
High
250
140
70
460
Satisfaction level
Medium
140
55
25
220
Low
10
5
5
20
TOTAL
400
200
100
700
A customer is chosen at random. Given that the customer uses the company more than five times per month,
what is the probability that the customer expressed medium satisfaction with the company?
1
5
1
59
A)
B)
C)
D)
4
44
28
140
8) Each manager of a corporation was rated as being either a good, fair, or poor manager by his/her boss. The
manager's educational background was also noted. The data appear below:
Educational Background
Manager
Rating H. S. Degree Some College College Degree Master's or Ph.D. Totals
Good
1
5
28
5
39
Fair
4
13
47
23
87
Poor
6
2
9
17
34
Totals
11
20
84
45
160
Given that a manager is rated as fair, what is the probability that this manager has no college background?
4
4
1
47
A)
B)
C)
D)
87
11
40
80
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9) Four hundred accidents that occurred on a Saturday night were analyzed. The number of vehicles involved
and whether alcohol played a role in the accident were recorded. The results are shown below:
Number of Vehicles Involved
Did Alcohol Play a Role?
1
2
3 or more Totals
Yes
52
99
19
170
No
29
172
29
230
Totals
81
271
48
400
Given that an accident involved multiple vehicles, what is the probability that it involved alcohol?
118
59
19
19
A)
B)
C)
D)
319
200
48
400
10) A researcher investigated whether a student's seat preference was related in any way to the gender of the
student. The researcher divided a lecture room into three sections (1 front, middle of the room, 2 front, sides
of the classroom, and 3 back of the classroom, both middle and sides) and noted where each student sat on a
particular day of the class. The researcher's summary table is provided below.
Male
Female
Total
Area 1
16
13
29
Area 2
8
10
18
Area 3
9
16
25
Total
33
39
72
Suppose a person sitting in the front, middle portion of the class is randomly selected to answer a question.
Find the probability that the person selected is female.
13
1
29
13
A)
B)
C)
D)
29
3
39
72
11) The manager of a used car lot took inventory of the automobiles on his lot and constructed the following table
based on the age of each car and its make (foreign or domestic):
Make
Foreign
Domestic
Total
0
43
36
79
2
Age of Car (in years)
3 5
6 10
26
10
28
14
54
24
over 10
21
22
43
Total
100
100
200
A car was randomly selected from the lot. Given that the car selected was a foreign car, what is the probability
that it was older than 2 years old?
57
43
57
43
A)
B)
C)
D)
100
100
121
121
12) A sample of 350 students was selected and each was asked the make of their automobile (foreign or domestic)
and their year in college (freshman, sophomore, junior, or senior). The results are shown in the table below.
Given that you know the selected student is in the senior class, find the probability they drive a domestic
automobile.
A) 25/35
B) 10/35
C) 15/350
D) 15/205
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13) A medium sized company characterized their employees based on the sex of the employee and their length of
service to the company. The results are summarized in the table below.
Suppose an employee has been randomly selected from this company. Given that the employee is male, find
the probability that they have worked for the company for more than 10 years?
A) 20/130
B) 20/30
C) 20/65
D) 75/130
14) The table shows the political affiliations and types of jobs for workers in a particular state. Suppose a worker is
selected at random within the state and the worker's political affiliation and type of job are noted.
Political Affiliation
Republican
Democrat
Independent
White collar
18%
13%
6%
Type of job
Blue Collar
19%
14%
30%
Given that the worker is a Democrat, what is the probability that the worker has a white collar job.
A) 0.481
B) 0.351
C) 0.255
D) 0.529
15) A local country club has a membership of 600 and operates facilities that include an 18 hole championship golf
course and 12 tennis courts. Before deciding whether to accept new members, the club president would like to
know how many members regularly use each facility. A survey of the membership indicates that 57% regularly
use the golf course, 48% regularly use the tennis courts, and 9% use both of these facilities regularly. Given that
a randomly selected member uses the tennis courts regularly, find the probability that they also use the golf
course regularly.
A) .1875
B) .1343
C) .7164
D) .4737
16) For two events, A and B, P(A) .4, P(B)
A) .29
B) .5
17) For two events, A and B, P(A)
A)
1
2
1
, P(B)
2
B)
3
4
18) For two events, A and B, P(A) .6, P(B)
A) .4
B) .3
19) For two events, A and B, P(A)
A)
5
8
3
, P(B)
4
B)
5
9
.7, and P(A
B)
1
, and P(A
3
B)
.8, and P(A | B)
2
, and P(B | A)
3
.2. Find P(A | B).
C) .08
1
. Find P(B | A).
4
C)
1
8
.5. Find P(A
C) .625
5
. Find P(A
6
C)
1
2
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B).
D) .14
D)
1
12
D) .833
B).
D)
9
10
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
20) A clothing vendor estimates that 78 out of every 100 of its online customers do not live within 50 miles of one
of its physical stores. It further estimates that 39 out of every 100 of its online customers is a man who does not
live within 50 miles of one of its physical stores. Using this estimate, what is the probability that a randomly
selected online customer is a man given that the customer does not live within 50 miles of a physical store?
21) A fast food restaurant chain with 700 outlets in the United States has recorded the geographic location of its
restaurants in the accompanying table of percentages. One restaurant is to be chosen at random from the 700 to
test market a new chicken sandwich.
Region
NE
SE
SW
NW
10,000
3%
6%
3%
0%
Population of City 10,000 100,000
15%
10%
12%
5%
100,000
20%
4%
1%
21%
What is the probability that the restaurant is located in a city with a population over 100,000, given that it is
located in the southwestern United States?
22) The table shows the political affiliations and types of job for workers in a particular state. Suppose a worker is
selected at random within the state and the worker's political affiliation and type of job are noted.
Political Affiliation
Republican
Democrat
Independent
White collar
5%
10%
15%
Type of job
Blue Collar
19%
17%
34%
Given that a worker is a blue collar worker, what is the probability that the worker is a Democrat?
23) A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows.
A: {The number is even}
B: {The number is less than 7}
Find P(A | B) and P(B | A).
24) A pair of fair dice is tossed. Events A and B are defined as follows.
A: {The sum of the dice is 7}
B: {At least one of the numbers is 3}
Find P(A | B) and P(B | A).
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25) The table shows the number of each Ford car sold in the United States in June 2006. Suppose the sales record
for one of these cars is randomly selected and the type of car is identified.
Type of Car
Sedan
Convertible
Wagon
SUV
Van
Hatchback
Total
Number
7,204
9,089
20,418
13,691
15,837
15,350
81,589
Events A and B are defined as follows.
A: {Convertible, SUV, Van}
B: {Fewer than 10,000 of the type of car were sold in June 2006}
Find P(A | B) and P(B | A).
26) Suppose that 62% of the employees at a company are male and that 35% of the employees just received merit
raises. If 20% of the employees are male and received a merit raise, what is the probability that a randomly
chosen employee is male given that the employee received a merit raise?
27) The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Suppose
that the die is rolled once. Let A be the event that the number rolled is less than 4, and let B be the event that the
number rolled is odd.
Outcome
Probability
1
.1
Find P(A | B).
2
.1
3
.1
4
.2
5
.2
6
.3
28) The data below show the types of medals won by athletes representing the United States in the Winter
Olympics. Suppose that one medal is chosen at random and the type of medal noted.
gold
bronze
gold
gold
gold
gold
silver
gold
silver
silver
silver
bronze
gold
silver
bronze
bronze
bronze
bronze
bronze
silver
silver
gold
silver
gold
silver
Given that the medal is not bronze, what is the probability that the medal is gold?
29) The data show the total number of medals (gold, silver, and bronze) won by each country winning at least one
gold medal in the Winter Olympics.
1
11
2
14
3
14
3
19
4
22
9
23
9
24
11
25
11
29
Suppose that one of the countries represented is selected at random and the total number of medals won by
that country is noted. What is the probability that the country won at least 25 medals given that the country did
not win fewer than 10 medals?
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Answer the question True or False.
30) The conditional probability of event A given that event B has occurred is written as P(B | A).
A) True
B) False
31) If A and B are mutually exclusive events, then P(A | B)
A) True
0.
B) False
32) For all events A and B, the conditional probabilities P(A | B) and P(B | A) are equal.
A) True
B) False
33) If every sample point in event B is also a sample point in event A, then P(A | B)
A) True
B) False
34) For any events A and B, P(A | B)
occur.
A) True
35) For any events A and B, P(A | B)
occur.
A) True
1.
P(Ac | B)
1, meaning given that B occurs either A occurs or A does not
P(A | Bc)
1, meaning given that A occurs either B occurs or B does not
B) False
B) False
3.6 The Multiplicative Rule and Independent Events
1 Use Multiplication Rule to Find Probability
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Suppose that for a certain experiment P(B)
A) 0.1
B) 0.7
2) Suppose that for a certain experiment P(A)
P(A B).
A) 0.18
B) 0.90
0.5 and P(A B)
0.6 and P(B)
0.2. Find P(A
C) 0.4
B).
D) 0.3
0.3. If A and B are independent events, find
C) 0.30
D) 0.50
3) A human gene carries a certain disease from a mother to her child with a probability rate of 0.47. That is, there
is a 47% chance that the child becomes infected with the disease. Suppose a female carrier of the gene has five
children. Assume that the infections, or lack thereof, are independent of one another. Find the probability that
all five of the children get the disease from their mother.
A) 0.023
B) 0.977
C) 0.042
D) 0.037
4) A machine has four components, A, B, C, and D, set up in such a manner that all four parts must work for the
machine to work properly. Assume the probability of one part working does not depend on the functionality of
any of the other parts. Also assume that the probabilities of the individual parts working are P(A) P(B) 0.95,
P(C) 0.91, and P(D) 0.98. Find the probability that the machine works properly.
A) 0.8048
B) 0.8472
C) 0.8213
D) 0.1952
5) Suppose a basketball player is an excellent free throw shooter and makes 92% of his free throws (i.e., he has a
92% chance of making a single free throw). Assume that free throw shots are independent of one another. Find
the probability that the player will make three consecutive free throws.
A) 0.7787
B) 0.2213
C) 0.0005
D) 0.9995
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6) Suppose a basketball player is an excellent free throw shooter and makes 96% of his free throws (i.e., he has a
96% chance of making a single free throw). Assume that free throw shots are independent of one another. Find
the probability that the player misses three consecutive free throws.
A) 0.0001
B) 0.1153
C) 0.8847
D) 0.9999
7) A one week study revealed that 60% of a warehouse store’s customers are women and that 30% of women
customers spend at least $250 on a single visit to the store. Find the probability that a randomly chosen
customer will be a woman who spends at least $250.
A) 0.18
B) 0.90
C) 0.50
D) 0.36
8) A study revealed that 45% of college freshmen are male and that 18% of male freshmen earned college credits
while still in high school. Find the probability that a randomly chosen college freshman will be male and have
earned college credits while still in high school.
A) 0.081
B) 0.530
C) 0.400
D) 0.027
9) In a particular town, 20% of the homes have monitored security systems. If an alarm is triggered, the security
system company will contact the local police to alert them of the alarm. Of all the alarm calls that the local
police receive, they only have the manpower to answer 30% of the calls. Suppose we randomly sample one
home that was broken into over the last month from this town. What is the probability that this home has a
monitored security system and that the police answered the alarm call?
A) 0.2000
B) 0.3000
C) 0.0600
D) 0.9400
10) A basketball player has an 80% chance of making the first free throw he shoots. If he makes the first
free throw shot, then he has a 90% chance of making the second free throw he shoots. If he misses the first
free throw shot, then he only has a 70% chance of making the second free throw he shoots. Suppose this
player has been awarded two free throw shots. Find the probability that he makes at least one of the two shots.
A) 0.72
B) 0.94
C) 0.86
D) 0.80
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
11) Suppose that for a certain experiment P(A)
.15 and P(B A)
12) Suppose that for a certain experiment P(A)
P(A B).
.32 and P(B)
.8. Find P(A
B).
.55. If A and B are independent events, find
13) A human gene carries a certain disease from a mother to her child with a probability rate of 0.30. That is, there
is a 30% chance that the child becomes infected with the disease. Suppose a female carrier of the gene has four
children. Assume that the infections, or lack thereof, are independent of one another. Find the probability that
none of the children get the disease from their mother.
14) Suppose there is a 38% chance that a risky stock investment will end up in a total loss of your investment.
Because the rewards are so high, you decide to invest in three independent risky stocks. What is the probability
that all three stocks end up in total losses?
15) An exit poll during a recent election revealed that 55% of those voting were women and that 65% of the women
voting favored Democratic candidates. What is the probability that a randomly chosen participant of the exit
poll would be a woman who favored Democratic candidates?
16) In the game of Parcheesi each player rolls a pair of dice on each turn. In order to begin the game, you must roll
a five on at least one die, or a total of five on both dice. Find the probability that the player does not get to begin
the game on either the first or the second rolls.
17) If 80% of a website's visitors are teenagers and 60% of those teenaged visitors are male, find the percentage of
the website's visitors that are teenaged males.
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18) A certain game has a deck of numbered cards of various colors. The probability of drawing a green card from a
well shuffled deck is .25 and the probability of drawing a card numbered 3 is .1. Assuming that "green" and "3"
are independent events, find the probability of drawing a green card numbered 3.
2 Determine if Events are Independent
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) If P(A B) 0 and P(A) 0, then which statement is false?
A) Events A and B are independent.
B) Events A and B are dependent.
C) Events A and B are mutually exclusive.
D) Events A and B have no sample points in common.
2) A number between 1 and 10, inclusive, is randomly chosen. Events A, B, C, and D are defined as follows.
A: {The number is even}
B: {The number is less than 7}
C: {The number is less than or equal to 7}
D: {The number is 5}
Identify one pair of independent events.
A) A and B
B) A and C
C) A and D
D) B and D
3) The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Suppose
that the die is rolled once.
Outcome
Probability
1
.1
2
.1
3
.1
Events A, B, C, and D are defined as follows.
4
.2
5
.2
6
.3
A: {The number is even}
B: {The number is less than 4}
C: {The number is less than or equal to 5}
D: {The number is greater than or equal to 5}
Identify one pair of independent events.
A) A and D
B) A and B
4) If P(A) .55, P(B A)
A) .4
.4, P(A
C) B and C
D) B and D
B) .22, and A and B are independent events, find P(B).
B) .22
C) .55
D) .88
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
5) In a box of 50 markers, 30 markers are either red or black, 20 are missing their caps, and 12 markers are either
red or black and are missing their caps. Are the events "red or black" and "missing cap" dependent or
independent? Explain.
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6) A pair of fair dice is tossed. Events A and B are defined as follows.
A: {The sum of the numbers showing is odd}
B: {The sum of the numbers showing is 2, 11, or 12}
Are A and B independent events? Explain.
7) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red
chips. Events A and B are defined as follows.
A: {Both chips are the same color}
B: {At least one of the chips is blue}
Are A and B independent events? Explain.
8) At a certain university, 70% of the students own cars. However, only 45% of the residence hall students own
cars. Are the events owning a car and living in a residence hall independent? Explain.
9) On a certain statistics test, 20% of the students earned a score of 90 or above. It was also true that 20% of the
male students earned a score of 90 or above. Are the events earning a score of 90 or above and being male
independent? Explain.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
10) Classify the events as dependent or independent: Events A and B where P(A)
P(A and B) 0.81.
A) independent
B) dependent
11) Classify the events as dependent or independent: Events A and B where P(A)
P(A and B) 0.47.
A) dependent
B) independent
0.9, P(B)
0.9, and
0.6, P(B)
0.8, and
12) Suppose two dice, one blue and one red, are rolled and the outcomes of each are recorded. We define the
following two events:
A: sum of the roll is 7
B. the result of the blue die is a number greater than 4
Are the two events, A and B, independent events?
A) Yes
B) No
13) A basketball player has an 80% chance of making the first free throw he shoots. If he makes the first
free throw shot, then he has a 90% chance of making the second free throw he shoots. If he misses the first
free throw shot, then he only has a 70% chance of making the second free throw he shoots. Suppose this
player has been awarded two free throw shots. Are the events, A the player makes the first shot, and B the
player makes the second shot, independent events?
A) Yes
B) No
Answer the question True or False.
14) If A and B are independent events, then A and B are also mutually exclusive.
A) True
B) False
15) Two events, A and B, are independent if P(A and B)
A) True
P(A)
P(B).
B) False
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16) If A and B,are independent events, then P(A)
A) True
P(B A).
3.7 Random Sampling
B) False
1 Understand Random Sampling
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) From 8 names on a list, a sample of 3 will be asked about voting preferences in an upcoming election. How
many different samples are possible?
A) 56
B) 336
C) 6720
D) 168
2) To win at BIG GAME in a certain state, one must correctly select 6 numbers from a collection of 52 numbers
(1 through 52). The order in which the selections is made does not matter. How many different 6 number
selections are possible, assuming no repetition of numbers?
A) 20,361,111
B) 18,009,460
C) 312
D) 1.466 1010
3) Which of the following best characterizes a random sample?
A) Every set of n elements in the population has an equal probability of being chosen.
B) Methods are employed to guarantee that all subgroups of the population are represented in the sample.
C) All members of the sample voluntary provide all information requested in a timely manner.
D) No member of the sample is allowed to communicate with any other member of the sample.
4) The class roll for a summer section of a statistics class included 100 students. There were no students in this
class whose last name began with the letter "Z." Interestingly, every other letter of the alphabet (A through Y)
was represented by four student's last names. That is, there were four students with last names that began
with the letter "A", four that began with "B", etc. Suppose I want to randomly sample 25 of these 100 students.
Which of the following methods would result in a random sample?
A) Write the student names on slips of paper. Put all 100 slips of paper into a hat and select 25 slips of paper
out of the hat, mixing them up between selections.
B) Sample the first 25 students from the listing of all students.
C) Write the student names on slips of paper. Select the four slips of paper representing the students with
the last names that begin with "A." Put those four slips of paper into a hat, mix them together and pick
one of these names out of the hat. Do this for each of the 25 letters of the alphabet.
D) Sample the students with the longest last names, because long last names are cool.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
5) In BIG GAME, you must select six numbers from fifty two numbers to win the big prize. The numbers do not
have to be in a particular order. What is the probability that any particular six digit combination will be
chosen?
6) Describe how to use a random number generator to select a random sample from a population. Simulate
choosing a random sample of two people from the list of five people below as an example.
Terry
Blaine
Alicia
Portia
Judd
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3.8 Some Additional Counting Rules (Optional)
1 Use the Multiplicative Rule
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Determine the number of sample points in the sample space corresponding to the experiment of tossing a coin
6 times.
A) 64
B) 36
C) 12
D) 128
2) Determine the number of sample points in the sample space corresponding to the experiment of tossing 3 dice.
A) 216
B) 729
C) 18
D) 432
3) A financial advisor has recommended three high risk high yield investments and four low risk low yield
investments to one of her clients. In how many ways can the client select one of each type of investment for his
portfolio?
A) 12
B) 7
C) 81
D) 64
4) A rental company offers 12 choices of table linens, three choices of china, two choices of flatware, and three
choices of stemware. In how many ways can a customer choose one of each to set the tables for a wedding
reception?
A) 216
B) 20
C) 152
D) 108
5) Five people have collaborated on a new television show. Because they all have big egos, they each want their
names listed first in the credits for the new show. How many different ways can the names be ordered if they
are randomly listed?
A) 25
B) 3125
C) 120
D) 60
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
6) Determine the number of sample points in the sample space corresponding to the experiment of tossing two
coins and one die.
7) You need to fly from Dallas to Chicago. Five airlines offer direct flights, and each airline offers 4 direct flights
on the day you need to travel. How many ways can you choose a direct flight from Dallas to Chicago?.
8) A popular restaurant occasionally offers a three course meal for a fixed price. The customer chooses one
appetizer, one entree, and one dessert from a special menu. There are four choices for the appetizer, five
choices for the entree, and three choices for the dessert. How many different ways can a customer choose the
three courses?
9) A statistics quiz has 5 multiple choice questions with four answer choices each and 5 true/false questions. How
many different ways can a student answer the questions by guessing?
2 Use the Permutations Rule
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the numerical value.
5
1) P 4
A) 120
B) 5
C) 24
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D) 1
8
2) P 4
A) 1680
10
3) P 4
A) 5040
12
4) P 0
A) 1
Solve the problem.
B) 70
C) 2
D) 4
B) 210
C) 34
D) 6
B) 11
C) 12
D) 39,916,800
N
5) Which expression is equivalent to P n ?
A)
N!
(N n)!
B)
N!
n!
C)
6
6) Find the numerical value of P 4 .
A) 360
B) 30
5
7) Find the numerical value of P 5 .
A) 120
B) 25
N!
n!(N n)!
D)
N!n!
(N n)!
C) 15
D) 720
C) 3125
D) 1
8) A manager of a road construction company must send one engineer to each of three different job sites to meet
with government inspectors. There are 8 engineers available. How many different ways can the manager send
one engineer to each of the job sites? (Assume that sending a particular engineer to job site 1 is different than
sending the same engineer to either job site 2 or 3.)
A) 336
B) 6720
C) 56
D) 6
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
9) A consumer testing service is commissioned to rank the top 4 brands of dishwasher detergent. Ten brands are
to be included in the study. In how many different ways can the consumer testing service arrive at the ranking?
10) A panel of coaches ranks the high school football teams in a particular state each week. The panel unanimously
agrees on which five teams should comprise the top five but not on the order. In how many different ways can
the coaches rank the top five teams?
11) There are 15 candidates for three different executive positions. How many different ways could you fill the
positions?
12) Suppose there are four dangerous military missions, each requiring one soldier. In how many different ways
can 4 soldiers from a squadron of 100 be assigned to these four missions?
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3 Use the Partitions Rule
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) You have 4 tasks to complete and 10 people, including yourself, to complete the tasks. In how many ways can
you assign 4 people to the first task, 3 people to the second task, 2 people to the third task, and 1 person to the
fourth task?
A) 12,600
B) 151,200
C) 1024
D) 5040
2) A group of 12 people is traveling in three vehicles. If 4 people ride in each vehicle, in how many different ways
can the people be assigned to the vehicles?
A) 34,650
B) 11,550
C) 495
D) 11,880
3) Suppose you teach a class of twelve students that you wish to assign to three groups of four students each. In
how many different ways can groups be made?
A) 34,650
B) 369,600
C) 479,000,600
D) 1320
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
4) You have 5 tasks to complete and 15 people, including yourself, to complete the tasks. In how many ways can
you assign 3 people to each task?
5) A group of 13 people is traveling in three vehicles. If 6 people ride in the van, 5 people ride in the car, and 2
people ride in the pickup, in how many different ways can the people be assigned to the vehicles?
4 Use the Combinations Rule
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the numerical value.
9
1)
4
A) 126
B) 3024
C) 2
D) 120
A) 1
B) 0
C) 89
D) 90
A) 1
B) 0
C) 69
D) 70
A) 2415
B) 164,220
C) 483
D) 32,844
2)
90
90
3)
70
0
4)
70
68
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5 Use the Counting Rules to Compute Probability
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Your manager will assign her 9 employees to three tasks, 2 to task A, 4 to task B, and 3 to task C. If she
randomly assigns employees to tasks, what is the probability that you and your best friend will be the two
assigned to task A?
1
1
2
1
A)
B)
C)
D)
36
1260
9
512
2) You toss a coin 5 times. What is the probability of getting heads on the first toss and tails on the next four
tosses?
1
5
31
27
A)
B)
C)
D)
32
32
32
32
3) You toss a coin 4 times. What is the probability of getting exactly one head?
1
1
15
A)
B)
C)
4
16
16
4) You toss one coin and one die. What is the probability of getting heads and 3?
1
1
1
A)
B)
C)
12
6
4
D)
3
4
D)
1
8
5) You toss one coin and one die. What is the probability of getting heads and a number greater than 3?
1
1
1
1
A)
B)
C)
D)
6
12
4
8
6) A panel of coaches ranks the high school football teams in a particular state each week. The panel unanimously
agrees on which five teams should comprise the top five but not on the order. Riverside High is among the top
five. If the ranking of the top five is random, what is that probability that Riverside High will be ranked first?
1
1
1
1
A)
B)
C)
D)
120
24
32
5
7) A panel of coaches ranks the high school football teams in a particular state each week. The panel unanimously
agrees on which five teams should comprise the top five but not on the order. Hillside High is among the top
five. If the ranking of the top five is random, what is that probability that Hillside High will be ranked either
first or second?
2
1
2
1
A)
B)
C)
D)
5
60
15
16
8) A panel of coaches ranks the high school football teams in a particular state each week. The panel unanimously
agrees on which five teams should comprise the top five but not on the order. Riverside High and Hillside
High are among the top five. If the ranking of the top five is random, what is that probability that Riverside
High will be ranked first and Hillside High will be ranked second?
1
1
1
1
A)
B)
C)
D)
20
120
60
40
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9) A panel of coaches ranks the high school football teams in a particular state each week. The panel unanimously
agrees on which five teams should comprise the top five but not on the order. Riverside High and Hillside
High are among the top five. If the ranking of the top five is random, what is that probability that Riverside
High and Hillside High will be the top two?
1
1
1
1
A)
B)
C)
D)
10
120
60
20
10) There are 10 players in a tennis tournament and five courts available for the first round. If players are
randomly assigned to courts, what is the probability that the two most favored players are both assigned to the
main court?
1
1
1
1
A)
B)
C)
D)
113,400
22,680
1024
5
11) There are 10 players in a tennis tournament and five courts available for the first round. If players are
randomly assigned to courts, what is the probability that the two most favored players are both assigned to the
same court?
1
1
1
1
A)
B)
C)
D)
22,680
113,400
1024
5
12) There are 10 players in a tennis tournament and five courts available for the first round. If players are
randomly assigned to courts, what is the probability that the two most favored players are not assigned to the
same court?
22,679
113,399
1019
4
A)
B)
C)
D)
22,680
113,400
1024
5
13) Suppose six people apply for two managerial positions at a company — two men and four women. Because the
candidates are all considered equally qualified for the position, the company decides to randomly select the
two candidates who will get the positions. How many different ways can two people be selected from the six
applicants?
A) 720 ways
B) 360 ways
C) 30 ways
D) 15 ways
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14) You toss a coin five times. What is the probability of getting heads on the first two tosses and tails on the next
three tosses?
15) You toss one coin and one die. What is the probability that you get heads and a number greater than 2?
16) Store Brand A fabric softener is among 10 brands being rated by a consumer group. If the group ranks the
brands at random, what is the probability that Store Brand A fabric softener will be ranked first?
17) National Brand B fabric softener is among 10 brands being rated by a consumer group. If the group ranks the
brands at random, what is the probability that National Brand B fabric softener will be ranked in the top three?
18) Your manager will assign her 10 employees to three tasks, 5 to task A, 3 to task B, and 2 to task C. If she
randomly assigns employees to tasks, what is the probability that you and your best friend will be the two
assigned to task C?
19) Your manager will assign her 10 employees to three tasks, 5 to task A, 3 to task B, and 2 to task C. If she
randomly assigns employees to tasks, what is the probability that you and your best friend will be assigned to
the same task?
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3.9 Bayes's Rules (Optional)
1 Use Bayes's Rule
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Suppose that B1 and B2 are mutually exclusive and complementary events, such that P(B1 )
Consider another event A such that P(A | B1 ) .2 and P(A | B2 ) .5. Find P(A).
.6 and P(B2 )
A) .32
B) .70
C) .38
D) .88
A) .375
B) .625
C) .240
D) .800
2) Suppose that B1 and B2 are mutually exclusive and complementary events, such that P(B1 )
Consider another event A such that P(A | B1 ) .2 and P(A | B2 ) .5. Find P(B1 | A).
.6 and P(B2 )
.4.
.4.
3) Suppose the probability of an athlete taking a certain illegal steroid is 10%. A test has been developed to detect
this type of steroid and will yield either a positive or negative result. Given that the athlete has taken this
steroid, the probability of a positive test result is 0.995. Given that the athlete has not taken this steroid, the
probability of a negative test result is 0.992. Given that a positive test result has been observed for an athlete,
what is the probability that they have taken this steroid?
A) 0.0995
B) 0.9928
C) 0.9552
D) 0.9325
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
4) An exit poll during a recent election revealed that 52% of those voting were women and 48% were men. The
results also showed that 70% of the women voting favored Democratic candidates while only 40% of the men
favored Democratic candidates. These poll results may be summarized as follows:
P(woman) .52
P(favored Democrats | woman) .70
a.
b.
c.
d.
e.
P(man) .48
P(favored Democrats | man)
Find P(woman and favored Democrats).
Find P(man and favored Democrats).
Find P(favored Democrats).
Find P(woman | favored Democrats).
Find P(man | favored Democrats).
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