Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
If a fair die is rolled three times, find the probability of getting an even number each time.
1)
A)
1
8
B)
1
3
C)
3
8
D)
0.6
E)
none of the above
2)
Two marbles are randomly drawn in succession without replacement from an urn that contains 10
red marbles and 10 green marbles.The probability that one marble is red and the other is green is
2)
A)
1
4.
B)
1
20 .
C)
5
19 .
D)
10
19 .
E)
1
2.
3)
Each question on a four–question multiple–choice examination has three choices, only one of which
is correct. By answering each question in a random fashion, the probability that exactly two
questions are answered correctly is
3)
A)
11
27 .
B)
1
2.
C)
2
27 .
D)
8
27 .
E)
1
6.
4)
In a certain town, 30% of eligible voters are Democrats, 40% are Republicans, and the rest are
Independents. In the last election, 20% of the Democrats, 10% of the Republicans, and 30% of the
Independents voted. If an eligible voter is chosen at random, what is the probability that he or she
is a Republican who voted?
4)
A)
0.04
B)
0.1
C)
0.06
D)
0.4
E)
0.12
5)
An urn contains 4 red and 3 yellow marbles. If two marbles are randomly drawn without
replacement, find the probability the second one is yellow, given that the first marble drawn is red.
5)
A)
2
7
B)
3
7
C)
3
4
D)
1
2
E)
none of the above
6)
The number of ways in which a student can answer an eight–question true–false examination is
6)
A)
40,320
B)
256
C)
8
D)
16
E)
64
7)
The winner of a contest can choose any two of six different prizes. How many choices are possible?
7)
A)
25
B)
20
C)
34
D)
15
E)
30
8)
Two cards are randomly drawn without replacement from a standard deck of 52 cards. Find the
probability that the second card is a heart.
8)
A)
2
3
B)
1
3
C)
12
13
D)
3
4
E)
1
4
9)
Urn I contains three green and four red marbles, and Urn II contains one green, two white, and two
red marbles. A marble is randomly drawn from Urn I and placed into Urn II. A marble is then
randomly drawn from Urn II. If it is red, what is the probability that a red marble was drawn from
Urn I?
9)
A)
1
3
B)
3
7
C)
2
3
D)
5
7
E)
1
2
10)
If a pair of dice are rolled, the probability that the sum of the numbers of dots appearing is 9 or 10 is
10)
A)
5
36 .
B)
1
18 .
C)
7
36 .
D)
1
6.
E)
1
9.
11)
After a production run, it was found that 10% of the units produced had a faulty weld and 5% had
both a defective paint job and a faulty weld. If a unit is randomly selected from this run and it has a
faulty weld, what is the probability that it also has a defective paint job?
11)
A)
1
5
B)
1
2
C)
1
10
D)
1
20
E)
1
3
12)
If the odds in favor of an event E are 2:7, find P(E‘).
12)
A)
3
5
B)
5
7
C)
7
9
D)
2
7
E)
2
9
13)
A student must select two courses in the liberal arts and three courses in the social sciences. There
are six liberal arts courses and ten social science courses, all of which are different, from which the
student may choose. How many selections are possible?
13)
A)
135
B)
21,600
C)
6
D)
1800
E)
750
14)
A building contractor can order lumber from 5 suppliers, bricks from 4 suppliers, and hardware
from 8 suppliers. If the contractor chooses one supplier for each item, how many choices are
available for the three items?
14)
A)
4,096,000
B)
160
C)
320
D)
17
E)
120
15)
The value of 9C2 is
15)
A)
18.
B)
5040.
C)
36.
D)
181,440.
E)
72.
16)
If S=1, 2, 3, 4, 5, 6, 7, 8 is a sample space of an experiment with events E=2, 4, 6, 8 and
F=4, 5, 6, 7, 8 , then EF‘ =
16)
A)
1, 2, 3, 5, 7 .
B)
4 .
C)
5, 7 .
D)
2 .
E)
1, 2, 3, 4, 6, 8 .
17)
An aptitude test is believed to have 80% accuracy. Find the probability that the test will be accurate
for at least one of the next two persons who take the test.
17)
A)
0.68
B)
0.96
C)
0.92
D)
0.84
E)
none of the above
18)
A sample space is partitioned by events E andF, where P(E) =1
3. Suppose that S is an event such
that P(SE) =1
4 and P(SF) =4
5.Find P(FS) .
18)
A)
5
32
B)
34
37
C)
11
13
D)
32
37
E)
5
13
19)
Two fair dice are rolled twice. Find the probability of getting a total of 6 on one of the rolls and a
total of 11 on the other one.
19)
A)
1
3
B)
5
648
C)
7
36
D)
5
324
E)
none of the above
20)
From an ordinary deck of 52 playing cards, a two–card hand is dealt. In how many ways can the
two cards be of the same face value?
20)
A)
104
B)
1326
C)
2
D)
6
E)
78
21)
From a group of ten people, five are assigned to room A and two to room B. In how many ways can
the assignment be made?
21)
A)
15,120
B)
45
C)
10
D)
252
E)
2520
22)
If S=1, 2, 3, 4, 5, 6, 7, 8 is a sample space of an experiment with events E=2, 4, 6, 8 and
F=4, 5, 6, 7, 8 , then EF‘ =
22)
A)
4 .
B)
2 .
C)
1, 2, 3, 5, 7 .
D)
1, 2, 3, 4, 6, 8 .
E)
5, 7 .
23)
The value of the product 4C1·3P3 is
23)
A)
6.
B)
3.
C)
24.
D)
4.
E)
12.
24)
The value of the sum 5P1+4C4 is
24)
A)
6.
B)
0.
C)
8.
D)
12.
E)
4.
25)
The value of 10C8 is
25)
A)
45.
B)
5
4.
C)
90.
D)
36.
E)
40.
26)
If P(E) = 0.4, P(EF) = 0.6, and P(E
F) = 0.1, find P(F’).
26)
A)
0.5
B)
0.6
C)
0.7
D)
0.3
E)
0.4
27)
The value of 10P3 is
27)
A)
120
B)
75,600
C)
30
D)
720
E)
70
28)
Three cards are randomly drawn, with replacement, from a standard deck of 52 cards. Find the
probability that the cards chosen, in order, are a queen, the 3 of diamonds, and a diamond.
28)
A)
3
8788
B)
12
7341
C)
2
5525
D)
3
8500
E)
none of the above
29)
If a pair of dice are rolled, the probability that the sum of the numbers of dots appearing is 5 is
29)
A)
1
18 .
B)
5
36 .
C)
1
3.
D)
1
9.
E)
1
12 .
30)
On Tuesday a store advertised a special sale for Wednesday. On Wednesday it was found that 60%
of customers had known about the sale and, of these customers, 40% bought a sale item. Of the
customers that had not known about the sale, 20% bought a sale item. If a customer bought a sale
item, what is the probability that he or she had known about the sale?
30)
A)
6
7
B)
3
4
C)
11
12
D)
7
8
E)
2
3
31)
How many distinguishable horizontal arrangements of all the letters in LETTERS are possible?
31)
A)
120
B)
1260
C)
21
D)
5040
E)
2520
32)
If a pair of dice are rolled, the probability that the sum of the numbers of dots appearing is not 4 is
32)
A)
7
12 .
B)
11
12 .
C)
2
3.
D)
1
9.
E)
8
9.
33)
Suppose you are in a class of 20 students and the instructor randomly calls on two students to give
reports. What is the probability that you will be called?
33)
A)
1
190
B)
1
10
C)
1
19
D)
1
20
E)
1
95
34)
A club has ten members. In how many ways can the offices of president, vice president, secretary,
and treasurer be filled if no member can serve in two offices?
34)
A)
5040
B)
10,000
C)
40
D)
16
E)
210
35)
A manufacturer of widgets has three assembly lines: A, B, and C. The percentages of total daily
output that are produced by the lines are 50%, 30%, and 20%, respectively. The percentages of
defective units produced by the lines are estimated to be 2%, 3%, and 1%, respectively. If a widget
is randomly selected from a day’s production and is defective, what is the probability that it came
from assembly line B?
35)
A)
3
7
B)
4
7
C)
8
21
D)
10
21
E)
2
7
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
A plane flies from New York to Chicago, from Chicago to Denver, and from Denver to Los Angeles. The plane’s on–time,
late, or early departure from each city is recorded as 0 or – or +
, respectively. For example, (+
, –, 0) represents the plane
leaving earlier than its scheduled departure time from New York, later than its scheduled departure time from Chicago,
and on–time from Denver. Assume that the time ahead of schedule or behind schedule is the same so that the pilot can
make up for a late departure. For example, (–, 0, +
) means the plane was x minutes late out of New York, on–time out of
Chicago, and made up the x minutes late out of New York by leaving Denver x minutes early. The plane is, therefore, on
time when it arrives in Los Angeles. Find the requested probability.
36)
The plane leaves on time in Chicago, given that it was late leaving New York.
36)
Provide an appropriate response.
37)
If P(E) = 0.3, P(F) = 0.4, and P(EF)= 0.2, find P(EF).
37)
38)
If S=a, b, c, d, e, f, g is a sample space of an experiment with events E=a, b, c ,
F=e, f, g , and G=a, c, d, g , find
(a) EF
(b) FG
(c) (E G)’
(d) E’
G
(e) Of the events E, F, and G, which pairs are mutually exclusive?
38)
39)
How many distinguishable arrangements of all letters in the word MISSISSIPPI are
possible
39)
40)
A family has two children. Determine
(a) the event E1 that at least one child is a boy;
(b) the event E2 that at least one child is a girl.
(c) Are E1 and E2 mutually exclusive?
40)
41)
A certain state has license plates that consist of 3 letters followed by 4 numerals. How
many different types of license plate can be made?
41)
42)
A United Nations diplomat can travel from Washington, D.C. to New York City in 6 ways.
Depending on the urgency of the trip, she can fly by helicopter, fly by plane, drive the car
to a suburban subway station and then take the subway the rest of the way, drive by
limousine, take the train, or borrow a friend’s yacht and sail most of the way. When she
returns to Washington, D.C., the same modes of transportation are available. How many
round trip transportation combinations are available to her?
42)
43)
If P(EF) =1
2, P(EF) =9
10 , and P(EF) =2
5, determine if E and F are independent or
dependent.
43)
44)
A man has a $1, a $5, a $10, and a $20 bill in his billfold. He also has a penny, a nickel, a
dime, and a quarter in his pants pocket. He decides to treat himself by spending one bill
and one coin. Determine the following events.
(a) E1= { the amount of money the man is prepared to spend, expressed in cents, is a
prime number}
(b) E2= {the amount of money the man is prepared to spend is more than one third of all
the money he is carrying}
(c) E3= {the man spends the nickel}
(d) E4= {the amount to be spent is one half the amount to be spent under another
selection}
(e) E‘2
(f) E3E4
(g) E3E4
(h) E1E4
(i) E1E‘1
(j) E1E‘1
44)
45)
A grocery store displays 100 green peppers in 10 rows of 10 each. It is quite likely that
customer A’s choice of the 2 best peppers to buy would differ from customer B’s choice,
even if the peppers were more or less identical. In how many ways (sample points) could
the first customer to see the green pepper array choose 2 peppers to buy?
45)
46)
An urn contains four marbles, numbers 1, 2, 3, and 4. If a marble is drawn and then a coin
is tossed, and the results are observed, determine the sample space of this experiment.
46)
47)
Determine the value of 6P2 and simplify your answer.
47)
48)
Urn I contains two red and three white marbles, and Urn II contains three red and four
white marbles. A marble is randomly drawn from Urn I and placed into Urn II. A marble is
then randomly drawn from Urn II. Find the probability that it is red.
48)
49)
Determine 10P7
49)
50)
The probability that Bob survives ten more years is 4
5, and the probability that Mary
survives ten more years is 5
6. Find the probability that exactly one of them survives ten
more years. (Assume independence.)
50)
51)
If the probability that a certain horse wins a race is 1
4, find the odds that this event occurs.
51)
52)
In a 20–question examination, each question is worth 5 points and is graded right or
wrong. Considering the individual questions, in how many ways can a student score 85
points or higher?
52)
53)
A college mathematics club of 30 students needs to elect new officers. If there are 3
positions available and no one can serve in more than one position, how many different
slates of candidates are possible?
53)
54)
How many distinguishable horizontal arrangements of all the letters in BOOKS are
possible?
54)
55)
A plane flies from New York to Chicago, from Chicago to Denver, and from Denver to Los
Angeles.The plane’s on–time, late, or early departure from each city is recorded as 0 or – or
+, respectively. For example, (+, –, 0) represents the plane leaving earlier than its scheduled
departure time from New York, later than its scheduled departure time from Chicago, and
on–time from Denver. Assume that the time ahead of schedule or behind schedule is the
same so that the pilot can make up for a late departure. For example, (–, 0, +) means the
plane was x minutes late out of New York, on–time out of Chicago, and made up the x
minutes late out of New York by leaving Denver x minutes early. The plane is, therefore,
on time when it arrives in Los Angeles. Determine the following events:
(a) E1= {the plane arrives in L.A. on time}
(b) E2= {the plane is late on arrival in L.A.}
(c) E3= {the plane is on time when it arrives at Denver}
(d) E4= {the plane arrives in L.A. as late as possible}
(e) E‘2
(f) E3E4
(g) E‘2E3
(h) E‘2E4
(i) E2E‘2
(j) E2E‘2
55)
56)
An urn contains two red and three green marbles. Two marbles are randomly drawn in
succession without replacement. Determine the probability that
(a) the first marble is red and the second is green;
(b) both marbles are red.
56)
57)
Fourteen children from a day care center are the first to arrive at a movie theater that seats
600. They decide to pull a trick on their chaperons by hiding under the seats, one child per
seat. If we consider the 14 seats selected to be one “hiding place”, how many potential
hiding places would the chaperons conceivably have to search? (Leave answer in factorial
form.)
57)
58)
A math instructor has twelve students in his class. It so happens that he has exactly 12
desks in his classroom. How many ways can the 12 students be arranged in the 12 desks?
58)
59)
The plane arrives late in L.A. but left Denver as scheduled.
59)
Provide an appropriate response.
60)
A car rental agency buys 140 new tires to outfit its 35–vehicle fleet. The tires can be
distinguished by their production serial numbers. Suppose the tire dealer delivers the tires
and dumps them in a heap. The rental agency’s maintenance person selects 4 tires from the
heap to put on the first car. How many ways can she select the 4 tires to put on the car?
60)
A plane flies from New York to Chicago, from Chicago to Denver, and from Denver to Los Angeles. The plane’s on–time,
late, or early departure from each city is recorded as 0 or – or +
, respectively. For example, (+
, –, 0) represents the plane
leaving earlier than its scheduled departure time from New York, later than its scheduled departure time from Chicago,
and on–time from Denver. Assume that the time ahead of schedule or behind schedule is the same so that the pilot can
make up for a late departure. For example, (–, 0, +
) means the plane was x minutes late out of New York, on–time out of
Chicago, and made up the x minutes late out of New York by leaving Denver x minutes early. The plane is, therefore, on
time when it arrives in Los Angeles. Find the requested probability.
61)
The plane is on time at L.A., when it left New York on time but left Chicago later than
scheduled.
61)
Provide an appropriate response.
62)
In a math course of 12 students, the instructor decides that he would like 4 students to go to
the board simultaneously to present 4 different problems to the class. How many ways can
he choose the 4 students?
62)
63)
A certain disease is believed to affect 4% of the population. Results of a new blood test for
the disease indicate that 95% of persons who have the disease will have a positive reaction
to the test, whereas 5% of those who do not have the disease will also have a positive
reaction. What is the probability that a randomly selected person who has a positive
reaction will actually have the disease?
63)
64)
In a certain class, 40% of students had a B average at midterm. Of these, 50% ended up
with a course grade of B. Of those who did not have a B average at midterm, 40% ended up
with a course grade of B. If one of the students in the class is selected at random and is
found to have received a B for the course, what is the probability that the student did not
have a B average at midterm?
64)
65)
Two cards are randomly drawn with replacement from a standard deck of 52 cards. Find
the probability of drawing, in order, the queen of hearts and a diamond.
65)
15
66)
A state legislative body is considering an action that allows gambling within the state. An
opinion survey of 200 voters was conducted and the results are indicated in the following
table:
Favor Oppose No Opinion Total
Democrat 40 35 984
Republican 60 30 696
Other 4 8 8 20
Total 104 73 23 200
Assume that the survey reflects the opinion of the voting population. If a person is selected
at random, determine each of the following (empirical) probabilities:
(a) P (opposes gambling)
(b) P (Republican)
(c) P (Democrat who favors gambling).
66)
67)
A youth sports team plays seven games. In how many ways can the outcomes of the
games result in five wins and two losses?
67)
68)
At a restaurant a complete dinner consists of a salad, an entree, a dessert, and a beverage.
For the salad, the choices are tossed green salad, gelatin salad, or cottage cheese; for the
entree, the choices are chicken, roast beef, or flounder; for the dessert, the choices are
pudding, pie, cake, or ice cream; for the beverage, the choices are coffee, tea, or milk. How
many complete dinners are possible?
68)
69)
In how many ways is it possible to answer a five–question multiple–choice examination if
each question has three choices and exactly one choice is selected for each question?
69)
70)
If a fair die is rolled three times, find the probability that a 3 or 5 comes up each time.
70)
16