Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
The number of ways in which a student can answer an eight–question true–false examination is
1)
A)
8
B)
256
C)
16
D)
40,320
E)
64
2)
If a pair of dice are rolled, the probability that the sum of the numbers of dots appearing is 5 is
2)
A)
1
12 .
B)
1
9.
C)
1
18 .
D)
1
3.
E)
5
36 .
3)
The value of 9C2 is
3)
A)
181,440.
B)
72.
C)
5040.
D)
18.
E)
36.
4)
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‘ =
4)
A)
2 .
B)
1, 2, 3, 5, 7 .
C)
1, 2, 3, 4, 6, 8 .
D)
4 .
E)
5, 7 .
5)
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
5)
A)
2
27 .
B)
11
27 .
C)
1
6.
D)
1
2.
E)
8
27 .
6)
If a pair of dice are rolled, the probability that the sum of the numbers of dots appearing is not 4 is
6)
A)
1
9.
B)
7
12 .
C)
2
3.
D)
8
9.
E)
11
12 .
7)
The value of 10C8 is
7)
A)
36.
B)
40.
C)
90.
D)
5
4.
E)
45.
8)
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
8)
A)
5
19 .
B)
1
4.
C)
1
20 .
D)
1
2.
E)
10
19 .
9)
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?
9)
A)
0.06
B)
0.04
C)
0.1
D)
0.4
E)
0.12
10)
Two cards are randomly drawn without replacement from a standard deck of 52 cards. Find the
probability that the second card is a heart.
10)
A)
3
4
B)
2
3
C)
1
3
D)
1
4
E)
12
13
11)
If a pair of dice are rolled, the probability that the sum of the numbers of dots appearing is 9 or 10 is
11)
A)
7
36 .
B)
5
36 .
C)
1
18 .
D)
1
6.
E)
1
9.
12)
The winner of a contest can choose any two of six different prizes. How many choices are possible?
12)
A)
30
B)
34
C)
15
D)
25
E)
20
13)
The value of the product 4C1·3P3 is
13)
A)
4.
B)
3.
C)
6.
D)
12.
E)
24.
14)
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?
14)
A)
3
7
B)
2
7
C)
4
7
D)
10
21
E)
8
21
15)
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.
15)
A)
0.68
B)
0.92
C)
0.96
D)
0.84
E)
none of the above
16)
The value of the sum 5P1+4C4 is
16)
A)
0.
B)
6.
C)
4.
D)
8.
E)
12.
17)
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?
17)
A)
1
95
B)
1
20
C)
1
10
D)
1
190
E)
1
19
18)
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.
18)
A)
3
8788
B)
2
5525
C)
3
8500
D)
12
7341
E)
none of the above
19)
The value of 10P3 is
19)
A)
70
B)
120
C)
30
D)
720
E)
75,600
20)
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?
20)
A)
10
B)
45
C)
2520
D)
15,120
E)
252
21)
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.
21)
A)
3
7
B)
3
4
C)
1
2
D)
2
7
E)
none of the above
22)
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?
22)
A)
2
3
B)
1
2
C)
5
7
D)
3
7
E)
1
3
23)
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?
23)
A)
1800
B)
750
C)
6
D)
21,600
E)
135
24)
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.
24)
A)
5
648
B)
1
3
C)
5
324
D)
7
36
E)
none of the above
25)
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) .
25)
A)
5
32
B)
34
37
C)
11
13
D)
5
13
E)
32
37
26)
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?
26)
A)
1
2
B)
1
3
C)
1
20
D)
1
5
E)
1
10
27)
How many distinguishable horizontal arrangements of all the letters in LETTERS are possible?
27)
A)
1260
B)
5040
C)
120
D)
2520
E)
21
28)
If a fair die is rolled three times, find the probability of getting an even number each time.
28)
A)
3
8
B)
0.6
C)
1
3
D)
1
8
E)
none of the above
29)
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‘ =
29)
A)
2 .
B)
1, 2, 3, 4, 6, 8 .
C)
5, 7 .
D)
1, 2, 3, 5, 7 .
E)
4 .
30)
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?
30)
A)
210
B)
16
C)
5040
D)
40
E)
10,000
31)
If P(E) = 0.4, P(EF) = 0.6, and P(E
F) = 0.1, find P(F’).
31)
A)
0.6
B)
0.7
C)
0.3
D)
0.5
E)
0.4
32)
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?
32)
A)
120
B)
160
C)
320
D)
4,096,000
E)
17
33)
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?
33)
A)
2
3
B)
6
7
C)
3
4
D)
7
8
E)
11
12
34)
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?
34)
A)
6
B)
1326
C)
78
D)
2
E)
104
35)
If the odds in favor of an event E are 2:7, find P(E‘).
35)
A)
3
5
B)
5
7
C)
2
9
D)
7
9
E)
2
7
36)
How many distinguishable horizontal arrangements of all the letters in BOOKS are
possible?
36)
37)
Three fair coins are tossed. Find the probability that
(a) three tails show;
(b) exactly two tails show.
37)
38)
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?
38)
39)
If P(E) =1
3, P(F‘) =2
5, and P(EF) =1
5, find P(EF).
39)
40)
In how many ways is it possible to answer a six–question true–false examination.
40)
41)
Determine the value of 5P3 and simplify your answer.
41)
42)
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?
42)
43)
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.)
43)
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.
44)
The plane arrives late in L.A. but left Denver as scheduled.
44)
45)
In a track race of 8 contestants, how many ways can the 1st, 2nd, and 3rd places finishes
happen?
45)
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.
46)
The plane leaves on time in Chicago, given that it was late leaving New York.
46)
Provide an appropriate response.
47)
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?
47)
48)
A sample space is partitioned by events E, F, and G, where P(E) =1
4, P(F) =1
2, and P(G) =
1
4. Suppose that S is an event such that P(SE) =3
5, P(SF) =3
5, AND P(SG) =1
5. Find P(E
S).
48)
49)
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).
49)
50)
If P(EF) =1
2, P(EF) =9
10 , and P(EF) =2
5, determine if E and F are independent or
dependent.
50)
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.
51)
The plane arrives on time in L.A. even though it left Denver later than scheduled.
51)
52)
Determine 8P5
52)
53)
Two fair dice are rolled. What is the probability that the sum of the dots appearing is
(a) 2;
(b) 7?
53)
54)
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?
54)
55)
If S=1, 2, 3, 4, 5, 6 is a sample space of an experiment with events E=1, 3, 5 ,
F4, 5, 6 , and G=2, 4, 6 , find
(a) EF
(b) EG
(c) FG‘
(d) Of the events E, F, and G, which pairs are mutually exclusive?
55)
56)
In how many ways can a basketball coach assign the five different positions to her
eight–member team if all members are equally qualified at all positions?
56)
57)
In a certain state, all drivers are given a driver’s license code consisting of 2 letters followed
by 6 numerals. How many different drivers license codes are possible?
57)
58)
A company will hire five people: three for the assembly department and two for the
finishing department. There are eight applicants who are equally qualified to work in each
department. In how many ways can the company fill the positions?
58)
59)
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 30%, 40%, and 30%, respectively. The
percentage of defective units produced by the lines are estimated to be 2%, 3%, and 2%,
respectively. If a widget is randomly selected from a day‘s production and it is defective,
what is the probability that it came from assembly line A?
59)
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. 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
60)
61)
In a math course of 12 students, the instructor decides that he would like 4 students to go to
the board to present 4 different problems to the class. The students go to the board one at a
time. No student goes to the board more than once. How many ways can he choose the 4
students?
61)
62)
If a basketball league has five teams, how many different end–of–the–season rankings are
possible? Assume that there are no ties.
62)
63)
A fair coin is tossed and then a fair die is rolled. Determine the probability that
(a) a head and an odd number show;
(b) a 2 or 4 shows.
63)
64)
A family is having a group picture taken to mail to all the relatives. Mother, Father, Son,
and Daughter are being photographed, but somebody has to work the camera, so only 3
people at a time can be photographed. Last year’s pictures looked “funny” for some reason,
so this year they decide to sit next to each other in various orders to see which composition
is best. Let the ordered 4–tuple (M, D, S, B) represent the photographer (Mom here), the
person on the left in the picture (Dad here), the person in the middle in the picture (the
Sister, in this case), and the person sitting on the right (Brother here). Determine the
following events:
(a) E1= {Dad, the tallest, sits in the middle}
(b) E2= {the photographer is male and Mom is on the left}
(c) E3= {there is only one female in the picture}
(d) E4= {Dad is to the immediate left of Mom}
(e) E‘3
(f) E2E4
(g) E‘1E2
(h) E2E4
(i) E2E‘2
(j) E2E‘2
64)