CHAPTER 4—INTRODUCTION TO PROBABILITY
MULTIPLE CHOICE
1. The probability of at least one head in two flips of a coin is
a.
0.33
b.
0.50
c.
0.75
d.
1.00
2. Revised probabilities of events based on additional information are
a.
joint probabilities
b.
posterior probabilities
c.
marginal probabilities
d.
complementary probabilities
3. Posterior probabilities are computed using
a.
the classical method
b.
Chebyshev’s theorem
c.
the empirical rule
d.
Bayes’ theorem
4. The complement of P(A | B) is
a.
P(AC | B)
b.
P(A | BC)
c.
P(B | A)
d.
P(A B)
5. An element of the sample space is
a.
an event
b.
an estimator
c.
a sample point
d.
an outlier
6. The probability of an intersection of two events is computed using the
a.
addition law
b.
subtraction law
c.
multiplication law
d.
division law
7. If A and B are mutually exclusive, then
a.
P(A) + P(B) = 0
b.
P(A) + P(B) = 1
c.
P(A B) = 0
d.
P(A B) = 1
8. Posterior probabilities are
a.
simple probabilities
b.
marginal probabilities
c.
joint probabilities
d.
conditional probabilities
9. The range of probability is
a.
any value larger than zero
b.
any value between minus infinity to plus infinity
c.
zero to one
d.
any value between -1 to 1
10. Since the sun must rise tomorrow, then the probability of the sun rising tomorrow is
a.
much larger than one
b.
zero
c.
infinity
d.
None of the other answers is correct.
11. Any process that generates well-defined outcomes is
a.
an event
b.
an experiment
c.
a sample point
d.
None of the other answers is correct.
12. In statistical experiments, each time the experiment is repeated
a.
the same outcome must occur
b.
the same outcome cannot occur again
c.
a different outcome may occur
d.
None of the other answers is correct.
13. Each individual outcome of an experiment is called
a.
the sample space
b.
a sample point
c.
an experiment
d.
an individual
14. A sample point refers to a(n)
a.
numerical measure of the likelihood of the occurrence of an event
b.
set of all possible experimental outcomes
c.
individual outcome of an experiment
d.
All of these answers are correct.
15. The set of all possible sample points (experimental outcomes) is called
a.
a sample
b.
an event
c.
the sample space
d.
a population
16. The sample space refers to
a.
any particular experimental outcome
b.
the sample size minus one
c.
the set of all possible experimental outcomes
d.
both any particular experimental outcome and the set of all possible experimental
outcomes are correct
17. An experiment consists of three steps. There are four possible results on the first step, three possible
results on the second step, and two possible results on the third step. The total number of experimental
outcomes is
a.
9
b.
14
c.
24
d.
36
18. An experiment consists of tossing 4 coins successively. The number of sample points in this
experiment is
a.
16
b.
8
c.
4
d.
2
19. A lottery is conducted using three urns. Each urn contains chips numbered from 0 to 9. One chip is
selected at random from each urn. The total number of sample points in the sample space is
a.
30
b.
100
c.
729
d.
1,000
20. Three applications for admission to a local university are checked to determine whether each applicant
is male or female. The number of sample points in this experiment is
a.
2
b.
4
c.
6
d.
8
21. Assume your favorite football team has 2 games left to finish the season. The outcome of each game
can be win, lose or tie. The number of possible outcomes is
a.
2
b.
4
c.
6
d.
None of the other answers is correct.
22. Each customer entering a department store will either buy or not buy some merchandise. An
experiment consists of following 3 customers and determining whether or not they purchase any
merchandise. The number of sample points in this experiment is
a.
2
b.
4
c.
6
d.
8
23. A graphical device used for enumerating sample points in a multiple-step experiment is a
a.
bar chart
b.
pie chart
c.
histogram
d.
None of the other answers is correct.
24. Of five letters (A, B, C, D, and E), two letters are to be selected at random. How many possible
selections are there?
a.
20
b.
7
c.
5!
d.
10
25. The “Top Three” at a racetrack consists of picking the correct order of the first three horses in a race. If
there are 10 horses in a particular race, how many “Top Three” outcomes are there?
a.
302,400
b.
720
c.
1,814,400
d.
10
26. When the assumption of equally likely outcomes is used to assign probability values, the method used
to assign probabilities is referred to as the
a.
relative frequency method
b.
subjective method
c.
probability method
d.
classical method
27. A method of assigning probabilities that assumes the experimental outcomes are equally likely is
referred to as the
a.
objective method
b.
classical method
c.
subjective method
d.
experimental method
28. When the results of experimentation or historical data are used to assign probability values, the method
used to assign probabilities is referred to as the
a.
relative frequency method
b.
subjective method
c.
classical method
d.
posterior method
29. A method of assigning probabilities based upon judgment is referred to as the
a.
relative method
b.
probability method
c.
classical method
d.
None of the other answers is correct.
30. Of the last 100 customers entering a computer shop, 25 have purchased a computer. If the classical
method for computing probability is used, the probability that the next customer will purchase a
computer is
a.
0.25
b.
0.50
c.
1.00
d.
0.75
31. The probability assigned to each experimental outcome must be
a.
any value larger than zero
b.
smaller than zero
c.
one
d.
between zero and one
32. An experiment consists of four outcomes with P(E1) = 0.2, P(E2) = 0.3, and P(E3) = 0.4. The
probability of outcome E4 is
a.
0.500
b.
0.024
c.
0.100
d.
0.900
33. A graphical method of representing the sample points of a multiple-step experiment is
a.
a frequency polygon
b.
a histogram
c.
an ogive
d.
a tree diagram
34. A(n) __________ is a graphical representation in which the sample space is represented by a rectangle
and events are represented as circles.
a.
frequency polygon
b.
histogram
c.
Venn diagram
d.
tree diagram
35. A(n) __________ is a collection of sample points.
a.
probability
b.
permutation
c.
experiment
d.
event
36. Given that event E has a probability of 0.25, the probability of the complement of event E
a.
cannot be determined with the above information
b.
can have any value between zero and one
c.
must be 0.75
d.
is 0.25
37. The symbol shows the
a.
union of events
b.
intersection of events
c.
sum of the probabilities of events
d.
sample space
38. The union of events A and B is the event containing
a.
all the sample points common to both A and B
b.
all the sample points belonging to A or B
c.
all the sample points belonging to A or B or both
d.
all the sample points belonging to A or B, but not both
39. The probability of the union of two events with nonzero probabilities
a.
cannot be less than one
b.
cannot be one
c.
cannot be less than one and cannot be one
d.
None of the other answers is correct.
40. The symbol shows the
a.
union of events
b.
intersection of events
c.
sum of the probabilities of events
d.
None of the other answers is correct.
41. The addition law is potentially helpful when we are interested in computing the probability of
a.
independent events
b.
the intersection of two events
c.
the union of two events
d.
conditional events
42. If P(A) = 0.38, P(B) = 0.83, and P(A B) = 0.57; then P(A B) =
a.
1.21
b.
0.64
c.
0.78
d.
1.78
43. If P(A) = 0.62, P(B) = 0.47, and P(A B) = 0.88; then P(A B) =
a.
0.2914
b.
1.9700
c.
0.6700
d.
0.2100
44. If P(A) = 0.85, P(A B) = 0.72, and P(A B) = 0.66, then P(B) =
a.
0.15
b.
0.53
c.
0.28
d.
0.15
45. Two events are mutually exclusive if
a.
the probability of their intersection is 1
b.
they have no sample points in common
c.
the probability of their intersection is 0.5
d.
the probability of their intersection is 1 and they have no sample points in common
46. Events that have no sample points in common are
a.
independent events
b.
posterior events
c.
mutually exclusive events
d.
complements
47. The probability of the intersection of two mutually exclusive events
a.
can be any value between 0 to 1
b.
must always be equal to 1
c.
must always be equal to 0
d.
can be any positive value
48. If two events are mutually exclusive, then the probability of their intersection
a.
will be equal to zero
b.
can have any value larger than zero
c.
must be larger than zero, but less than one
d.
will be one
49. Two events, A and B, are mutually exclusive and each has a nonzero probability. If event A is known
to occur, the probability of the occurrence of event B is
a.
one
b.
any positive value
c.
zero
d.
any value between 0 to 1
50. If A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.5, then P(A B) =
a.
0.30
b.
0.15
c.
0.00
d.
0.20
51. If A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.5, then P(A B) =
a.
0.00
b.
0.15
c.
0.8
d.
0.2
52. In an experiment, events A and B are mutually exclusive. If P(A) = 0.6, then the probability of B
a.
cannot be larger than 0.4
b.
can be any value greater than 0.6
c.
can be any value between 0 to 1
d.
cannot be determined with the information given
53. Which of the following statements is(are) always true?
a.
-1 P(Ei) 1
b.
P(A) = 1 − P(Ac)
c.
P(A) + P(B) = 1
d.
both P(A) = 1 − P(Ac) and P(A) + P(B) = 1
54. One of the basic requirements of probability is
a.
for each experimental outcome Ei, we must have P(Ei) 1
b.
P(A) = P(Ac) − 1
c.
if there are k experimental outcomes, then P(E1) + P(E2) + … + P(Ek) = 1
d.
both P(A) = P(Ac) − 1 and if there are k experimental outcomes, then P(E1) + P(E2) + … +
P(Ek) = 1
55. Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.2. The probability of the
complement of Event B equals
a.
0.00
b.
0.06
c.
0.7
d.
None of the other answers is correct.
56. The multiplication law is potentially helpful when we are interested in computing the probability of
a.
mutually exclusive events
b.
the intersection of two events
c.
the union of two events
d.
None of the other answers is correct.
57. If P(A) = 0.80, P(B) = 0.65, and P(A B) = 0.78, then P(BA) =
a.
0.6700
b.
0.8375
c.
0.9750
d.
Not enough information is given to answer this question.
58. If two events are independent, then
a.
they must be mutually exclusive
b.
the sum of their probabilities must be equal to one
c.
the probability of their intersection must be zero
d.
None of the other answers is correct.
59. If A and B are independent events with P(A) = 0.38 and P(B) = 0.55, then P(AB) =
a.
0.209
b.
0.000
c.
0.550
d.
None of the other answers is correct.
60. If X and Y are mutually exclusive events with P(X) = 0.295, P(Y) = 0.32, then P(XY) =
a.
0.0944
b.
0.6150
c.
1.0000
d.
0.0000
61. Two events with nonzero probabilities
a.
can be both mutually exclusive and independent
b.
cannot be both mutually exclusive and independent
c.
are always mutually exclusive
d.
cannot be both mutually exclusive and independent and are always mutually exclusive
62. If P(A) = 0.50, P(B) = 0.60, and P(A B) = 0.30; then events A and B are
a.
mutually exclusive events
b.
not independent events
c.
independent events
d.
Not enough information is given to answer this question.
63. On a December day, the probability of snow is .30. The probability of a “cold” day is .50. The
probability of snow and a “cold” day is .15. Are snow and “cold” weather independent events?
a.
only if given that it snowed
b.
no
c.
yes
d.
only when they are also mutually exclusive
64. If P(A) = 0.5 and P(B) = 0.5, then P(A B) is
a.
0.00
b.
0.25
c.
1.00
d.
cannot be determined from the information given
65. If A and B are independent events with P(A) = 0.4 and P(B) = 0.6, then P(A B) =
a.
0.76
b.
1.00
c.
0.24
d.
0.2
66. If A and B are independent events with P(A) = 0.2 and P(B) = 0.6, then P(A B) =
a.
0.62
b.
0.12
c.
0.60
d.
0.68
67. If A and B are independent events with P(A) = 0.4 and P(B) = 0.25, then P(A B) =
a.
0.65
b.
0.55
c.
0.10
d.
Not enough information is given to answer this question.
68. Events A and B are mutually exclusive. Which of the following statements is also true?
a.
A and B are also independent.
b.
P(A B) = P(A)P(B)
c.
P(A B) = P(A) + P(B)
d.
P(A B) = P(A) + P(B)
69. If A and B are independent events with P(A) = 0.05 and P(B) = 0.65, then P(AB) =
a.
0.05
b.
0.0325
c.
0.65
d.
0.8
70. A six-sided die is tossed 3 times. The probability of observing three ones in a row is
a.
1/3
b.
1/6
c.
1/27
d.
1/216
71. If a coin is tossed three times, the likelihood of obtaining three heads in a row is
a.
zero
b.
0.500
c.
0.875
d.
0.125
72. If a penny is tossed four times and comes up heads all four times, the probability of heads on the fifth
trial is
a.
zero
b.
1/32
c.
0.5
d.
larger than the probability of tails
73. If a penny is tossed three times and comes up heads all three times, the probability of heads on the
fourth trial is
a.
smaller than the probability of tails
b.
larger than the probability of tails
c.
1/16
d.
None of the other answers is correct.
74. A perfectly balanced coin is tossed 6 times and tails appears on all six tosses. Then, on the seventh trial
a.
tails cannot appear
b.
heads has a larger chance of appearing than tails
c.
tails has a better chance of appearing than heads
d.
None of the other answers is correct.
75. The probability of the occurrence of event A in an experiment is 1/3. If the experiment is performed 2
times and event A did not occur, then on the third trial event A
a.
must occur
b.
may occur
c.
could not occur
d.
has a 2/3 probability of occurring
76. Bayes’ theorem is used to compute
a.
the prior probabilities
b.
the union of events
c.
both the prior probabilities and the union of events
d.
the posterior probabilities
77. Initial estimates of the probabilities of events are known as
a.
sets
b.
posterior probabilities
c.
conditional probabilities
d.
prior probabilities
78.
A
B
C
D
E
1
Prior
Conditional
Joint
2
Event
Probability
Probability
Probability
3
A1
0.25
0.31
For the Excel worksheet above, which of the following formulas would correctly calculate the joint
probability for cell D3?
a.
=SUM(B3:C3)
b.
B3+C3
c.
B3/C3
d.
=B3*C3
79.
A
B
C
D
E
1
Prior
Conditional
Joint
Posterior
2
Event
Probability
Probability
Probability
Probability
3
A1
0.45
0.22
0.099
4
A2
0.55
0.16
0.088
5
0.187
For the Excel worksheet above, which of the following formulas would correctly calculate the
posterior probability for cell E3?
a.
=SUM(B3:D3)
b.
=D3/$D$5
c.
=D5/$D$3
d.
B3/C3+D3
80. If P(A B) = 0,
a.
P(A) + P(B) = 1
b.
either P(A) = 0 or P(B) = 0
c.
A and B are mutually exclusive events
d.
A and B are independent events
81. The probability of an event is
a.
the sum of the probabilities of the sample points in the event
b.
the product of the probabilities of the sample points in the event
c.
the minimum of the probabilities of the sample points in the event
d.
the maximum of the probabilities of the sample points in the event
82. If P(A|B) = .3,
a.
P(B|A) = .7
b.
P(AC|B) = .7
c.
P(A|BC) = .7
d.
P(AC|BC) = .7
83. If A and B are independent events with P(A) = .1 and P(B) = .4, then
a.
P(A B) = 0
b.
P(A B) = .04
c.
P(A B) = .5
d.
P(A B) = .25
84. If P(A|B) = .3 and P(B) = .8, then
a.
P(A) = .24
b.
P(B|A) = .7
c.
P(A B) = .5
d.
P(A B) = .24
85. If P(A) = .6, P(B) = .3, and P(A B) = .2, then P(B|A) =
a.
.33
b.
.5
c.
.67
d.
.9
PROBLEM
1. All the employees of ABC Company are assigned ID numbers. The ID number consists of the first
letter of an employee’s last name, followed by four numbers.
a.
How many possible different ID numbers are there?
b.
How many possible different ID numbers are there for employees whose last name starts with
an “A”?
2. A company plans to interview 10 recent graduates for possible employment. The company has three
positions open. How many groups of three can the company select?
3. A student has to take 7 more courses before she can graduate. If none of the courses are prerequisites
to others, how many groups of three courses can she select for the next semester?
4. A committee of 4 is to be selected from a group of 12 people. How many possible committees can be
selected?
5. The sales records of a real estate agency show the following sales over the past 200 days:
Number of
Number
Houses Sold
of Days
0
60
1
80
2
40
3
16
4
4
a.
How many sample points are there?
b.
Assign probabilities to the sample points and show their values.
c.
What is the probability that the agency will not sell any houses in a given day?
d.
What is the probability of selling at least 2 houses?
a.
260,000
b.
10,000
e.
What is the probability of selling 1 or 2 houses?
f.
What is the probability of selling less than 3 houses?
6. The results of a survey of 800 married couples and the number of children they had is shown below.
Number
of Children
Probability
0
0.050
1
0.125
2
0.600
3
0.150
4
0.050
5
0.025
If a couple is selected at random, what is the probability that the couple will have
a.
Less than 4 children?
b.
More than 2 children?
c.
Either 2 or 3 children?
a.
0.925
b.
0.225
c.
0.75
7. An experiment consists of throwing two six-sided dice and observing the number of spots on the upper
faces. Determine the probability that
a.
the sum of the spots is 3.
b.
each die shows four or more spots.
c.
the sum of the spots is not 3.
d.
neither a one nor a six appear on each die.
e.
a pair of sixes appears.
f.
the sum of the spots is 7.
a.
5
b.
Probability
0
1
2
3
4
c.
0.3
d.
0.3
f.
0.9
8. Assume that in your hand you hold an ordinary six-sided die and a dime. You toss both the die and the
dime on a table.
a.
What is the probability that a head appears on the dime and a six on the die?
b.
What is the probability that a tail appears on the dime and any number more than 3 on the die?
c.
What is the probability that a number larger than 2 appears on the die?
a.
1/12
b.
3/12
c.
8/12
9. A very short quiz has one multiple-choice question with five possible choices (a, b, c, d, and e) and
one true or false question. Assume you are taking the quiz but do not have any idea what the correct
answer is to either question, but you mark an answer anyway.
a.
What is the probability that you have given the correct answer to both questions?
b.
What is the probability that only one of the two answers is correct?
c.
What is the probability that neither answer is correct?
d.
What is the probability that only your answer to the multiple-choice question is correct?
e.
What is the probability that you have only answered the true or false question correctly?
a.
1/10
b.
5/10
c.
4/10
d.
1/10
e.
4/10
10. Two of the cylinders in an eight-cylinder car are defective and need to be replaced. If two cylinders are
selected at random, what is the probability that
a.
both defective cylinders are selected?
b.
no defective cylinder is selected?
c.
at least one defective cylinder is selected?
a.
2/56
b.
30/56
c.
26/56
a.
2/36
b.
9/36
c.
34/36
d.
16/36
e.
1/36
6/36