71)
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?
71)
72)
If three cards are randomly drawn without replacement from a standard deck of 52 cards,
find the probability that all are queens.
72)
73)
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?
73)
74)
Find the probability of rolling the same number in four throws of a fair die.
74)
75)
An urn contains five chips numbered from 1 to 5. A chip is randomly drawn. Let E be the
event of drawing a 3 and F be the event of drawing a 5. Are E and F independent?
75)
76)
Determine 8C3
76)
77)
If a basketball league has five teams, how many different end–of–the–season rankings are
possible? Assume that there are no ties.
77)
78)
A family has 3 children, 2 girls and a boy. Every morning (Tuesday through Saturday) each
child selects a stock at random from the list printed in the newspaper. If the stock is up,
that child does the morning dishes. If the stock is down, the child who picked the stock
does the noon dishes. If the stock is unchanged, that child does the evening dishes. If there
is a tie–2 children to do the noon dishes, for example–then both children work together to
do the noon dishes, and the remaining meal’s dishes are done by default by mom or dad.
The assignments can be thought of as an ordered triple, ordered by age. For example, (+, 0,
–) means that the youngest does the morning dishes (the stock was up), the oldest child
does the noon dishes (the stock was down), and the middle child, whose stock was
unchanged, does the dishes after the evening meal. (–, 0, 0) means the youngest does the
noon dishes, the other two work together on the dishes from the evening meal. The
morning dishes are done by mom and dad. Determine the following events:
78)
17
(a) E1= {the oldest and youngest work together doing the dishes; the middle child does
the dishes from a different meal}
(b) E2= {the youngest child has the day off}
(c) E3= {mom and dad do the dishes twice}
(d) E4= {the oldest child does the noon dishes alone}
(e) E‘2
(f) E3E4
(g) Let E5= {either the oldest or the youngest child does the morning dishes alone}
Let E6= {the middle child does the evening dishes alone}
E5E6
(h) E3E4
(i) E2E‘2
(j) E2E‘2
79)
If two cards are randomly drawn, without replacement, from a standard deck of 52 cards,
find the probability that the second card is not a jack or queen, given that the first card is a
jack or queen.
79)
80)
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).
80)
81)
An urn contains 3 green, 2 yellow, and 6 red 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.
81)
18
82)
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?
82)
83)
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?
83)
84)
A state has decided to issue personalized license plates with 3 letters on them. The original
idea was that these letters would represent the vehicle owner’s first, middle, and last
initials, respectively. Unfortunately, some 3 letter combinations are more colorful than
others and the state received requests for combinations that in no way resembled the
person’s initials. The Department of Motor Vehicles decided to withdraw inappropriate or
suggestive combinations. How long was the list of personalized plates that had to be
reviewed?
84)
85)
Determine the value of 4C2 and simplify your answer.
85)
86)
If P(E) =1
3, P(F‘) =2
5, and P(EF) =1
5, find P(EF).
86)
87)
In a survey of newspaper readers, it was found that 40% like the Gazette, 25% liked the
Bulletin, and 10% liked both. If a person in the survey is randomly selected, find the
probability that the person liked the Bulletin, given that he or she liked the Gazette.
87)
88)
How many different two–card hands can be dealt from an ordinary deck of 52 playing
cards?
88)
89)
If P(E) = 0.2, P(F) = 0.6, and P(EF) = 0.12, determine if E and F are independent or
dependent.
89)
90)
Two fair dice are rolled. What is the probability that the sum of the dots appearing is
(a) 2;
(b) 7?
90)
91)
An urn contains three red and two green marbles, and a second urn contains two red and
two green marbles. An urn is selected at random and a marble is randomly drawn from it.
The marble is green. What is the probability that it came from the first urn?
91)
92)
If a die is rolled and then a coin is tossed, and the results are observed, determine the
sample space of this experiment.
92)
93)
Ten tickets numbered from 1 to 10 are placed in a hat. If two tickets are randomly drawn
with replacement, find the probability the sum is 15.
93)
94)
Determine the value of 3P3·3C3 and simplify your answer.
94)
95)
Five different books are to be arranged horizontally on a bookshelf. (a) In how many ways
can this be done? (b) If two are mathematics books and three are accounting books, in how
many ways can all the books be arranged if the first two books are to be in mathematics?
95)
96)
An office worker spends each coffee break at the same deli. This deli has the same 8
varieties of coffee every day. To add variety and excitement to his or her day, the office
worker decides to randomly choose a coffee variety in the morning and then randomly
choose a variety in the afternoon. How many morning coffee – afternoon coffee choices are
there?
96)
97)
If events E and F are independent with P(E) = 0.3 and P(F) = 0.5, find P(E‘ F).
97)
98)
If a fair red die and a fair green die are rolled, find the probability that the sum is greater
than 8, given that a 4 shows on the red die.
98)
99)
Two light bulbs are selected from a box of 20 bulbs. For this experiment, how many sample
points are in the sample space?
99)
100)
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.
100)
101)
From a lot of ten computers, two are selected for extensive testing. In how many ways can
the selection be made?
101)
102)
Given the equiprobable sample space S=1, 2, ,3, 4, 5 and events E=1, 2, 4 and F=
1, 4, 5 , find P(E F).
102)
103)
Two cards are randomly drawn with replacement from a standard deck of 52 playing
cards. Find the probability that
(a) both cards are aces;
(b) the first card is red and the second card is a club;
(c) one card is red and the other is a club.
103)
21
104)
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?
104)
105)
An urn contains ten marbles numbered 1 through 10. If two marbles are randomly drawn
in succession without replacement, determine the probability that
(a) the first marble drawn shows 1 and the second shows 2;
(b) both show an odd number;
(c) at least one marble shows a number greater than 5.
105)
106)
In a track race of 8 contestants, how many ways can the 1st, 2nd, and 3rd places finishes
happen?
106)
107)
A manufacturer places a four–symbol code on each unit of a product. The first three
symbols are numbers with the first not 0, and the fourth symbol is a letter other than o.
How many codes are possible?
107)
108)
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?
108)
109)
In how many ways can a four–member committee be selected from a group of seven
people?
109)
110)
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 25%, 35%, and 40%, respectively. The
percentages of defective units produced by the lines are estimated to be 1%, 2%, and 1%,
respectively. If a widget is randomly selected from a day‘s production, what is the
probability that it is defective?
110)
111)
Determine 8P5
111)
112)
From a group of 12 people, in how many ways can the offices of chairperson, secretary,
and treasurer be filled by three different people?
112)
113)
A quiz contains four multiple–choice problems. Each problem has five choices for the
answer, but only one of them is correct. If a student randomly guesses the answer to each
problem, find the probability that the student gets exactly three correct answers.
113)
114)
A card is randomly drawn from an ordinary deck of 52 playing cards. What is the
probability that it is
(a) a king;
(b) not a heart;
(c) a 10 and not red.
114)
115)
An urn contains ten marbles numbered 1 through 10. If a marble is randomly selected from
the urn, determine the probability that it shows a number greater than or equal to 4.
115)
116)
Determine the value of 5P3 and simplify your answer.
116)
117)
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. How many spending possibilities does he have?
117)
118)
Determine the value of 10C8 and simplify your answer.
118)
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.
119)
The plane arrives on time in L.A. even though it left Denver later than scheduled.
119)
Provide an appropriate response.
120)
A coin is tossed three times. Determine
(a) the event E1 that exactly two heads occur;
(b) the event E2 that at least two heads occur;
(c) the event E3 that no head occurs.
120)
121)
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
121)
122)
From an urn that contains eight marbles numbered 1 through 8, three marbles are
randomly selected in succession and the results observed. Determine the number of sample
points in the sample space of this experiment if the marbles are selected
(a) with replacement and
(b) without replacement.
122)
123)
Three fair coins are tossed. Find the probability that
(a) three tails show;
(b) exactly two tails show.
123)
124)
From a committee of five men and four women, a subcommittee consisting of two men and
two women is to be formed. In how many ways can the subcommittee be chosen?
124)
125)
Determine the value of 5P1·5C3 and simplify your answer.
125)
126)
In how many ways is it possible to answer a six–question true–false examination.
126)
127)
Determine 14C10
127)
Answer Key
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Answer Key
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Answer Key
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Answer Key
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Answer Key
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Answer Key
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