Full Name :
Seat number :
Math 166:503
Fall 2015
Exam 2 - Sample problems
Instructions:
• Answer the questions in the spaces provided. If you run out of room for an
answer, continue on the back of the test.
• Parts I and II are Multiple Choice and True/False. The multiple choice problems
have only one correct answer. Please mark your answer clearly, especially if you
change your answer. Any ambiguous answers will be considered incorrect.
• Part III is work out problems. All necessary work should be shown for full credit.
Partial credit may be awarded based on correct work shown. Answers with no
work will not receive full credit. Please box your final answer.
• Round all numerical answers to two decimal places. If it is a problem with
percentages, then round to four decimal places.
On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic
work.
Signature:
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Part I: Multiple Choice
1. The following questions have only one correct solution.
(a) (5 points) Bryan High School holds their annual science fair. They award a first prize,
a second prize, a third prize, and three honorable mentions. If there are 65 entries, how
many different ways can these 6 prizes be awarded?
A. 82,598,880
B. 9,911,865,600
C. 5.9471 x 1010
D. None of the above
(b) (5 points) A clothing store has 5 identical red shirts, 3 identical yellow shirts, 4 identical
blue shirts and 7 identical pink shirts. They wish to hang them up on a rack. How many
distinguishable ways can they arrange the shirts?
A. 1,396,755,360
B. 1.2165 x 1017
C. 1.9165 x 1023
D. None of the above
(c) (5 points) Seven friends go to the movies. There are four females and three males. In how
many ways can they arrange themselves in a single row if they want to alternate gender?
A. 12
B. 144
C. 288
D. 1225
E. 5040
F. None of the above
(d) (5 points) For a certain heart operation, the probability of survival is 0.95. If 10 of these
operations are performed every week, what is the expected number of deaths due to this
operation?
A. 9.5
B. 0.5
C. 0.475
D. None of these
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(e) (5 points) In a hand of Black Jack, a player is dealt 2 cards out of a standard 52-card
deck. A hand is a “black jack” if exactly one card is an ace and the other card is either
a ten or a face card (jack, queen, king). What is the probability that a given hand is a
“black jack”?
A. 0.0237
B. 0.0385
C. 0.0473
D. 0.0483
E. None of the above
The heights of fifth grade children are normally distributed with average 56 inches and
standard deviation 3 inches.
Use the above information for the following two questions.
(f) (5 points) What is the probability that a fifth grader is between 48 inches (4 feet) and 60
inches (5 feet) tall?
A. 0.9050
B. 0.0564
C. 0.9088
D. 0.0038
E. None of the above
(g) (5 points) How tall must a fifth grader be to be in the top 10% of tallest fifth graders?
A. 52.1553 in.
B. 59.8447 in.
C. 62.3241 in.
D. 74.7669 in.
E. None of the above
(h) (5 points) A computer password must consist of 6 characters. The first 2 must be uppercase letters, the next 2 must be digits (0-9), and the last two must be lowercase letters.
How many passwords can be formed that start with a vowel?
A. 45,697,600
B. 38,025,000
C. 8,788,000
D. 7,312,500
E. None of the above
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(i) (5 points) The Powerball lottery is a game in which 59 numbered white balls (numbered
1-59) are placed in one drum and 35 numbered red balls (numbered 1-35) are placed in
another drum. Five white balls are drawn and one red ball. Participants pay $2 to guess
which five white balls and which red ball will be drawn (not in any particular order). They
win the jackpot if they guess all six balls correctly and win nothing otherwise. What does
the jackpot need to be for the game to be fair?
A. 1,628,433,534
B. 4.20536424 x 1010
C. 175,223,510
D. 350,447,020
E. None of the above
(j) (4 points) Suppose that in a certain country, the probability a person weighs over 160
lbs is 0.58. If a group of 40 people is randomly selected from this country, what is the
probability that more than 15 but fewer than 21 weigh less than 160 lbs?
A. 0.1859
B. 0.5401
C. 0.1765
D. None of the above
(k) (5 points) In how many ways can four couples be seated in a row of eight seats at a theater
if each couple is seated together?
A. 48
B. 384
C. 2520
D. None of the above
(l) (5 points) A committee of three is to be selected at random from a group of three senior
and four junior executives. What is the probability that the committee will have more
senior than junior executives?
A. 0.0237
B. 0.3426
C. 0.3714
D. None of the above
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(m) (2 points) A class consists of 2 freshmen, 5 sophomores, 9 juniors and 10 seniors. Students
are selected one at a time at random from this class until a sophomore is found. No student
gets selected more than once. Let X denote the number of students selected in one trial
of this experiment. X is
A. finite discrete
B. infinite discrete
C. continuous
(n) (2 points) A farmer plants 10 watermelon seeds. Let X denote the number of the seeds
that sprout. X is
A. finite discrete
B. infinite discrete
C. continuous
(o) (2 points) Let X denote the number of minutes that a person waits in line at a grocery
store check-out lane during the 5 o’clock hour. X is
A. finite discrete
B. infinite discrete
C. continuous
(p) (2 points) A bag contains 50 jelly beans of assorted colors: red, green, blue, yellow, orange,
and black. Jelly beans are drawn from the bag without replacement. The random variable
X is the number of jelly beans you draw until you get a red one. X is
A. finite discrete
B. infinite discrete
C. continuous
(q) (2 points) A pair of unfair, six-sided dice are rolled. The random variable X is assigned
the value of the sum of the uppermost faces. X is
A. finite discrete
B. infinite discrete
C. continuous
(r) (2 points) The random variable X is the number of minutes a student slept in a math
class on a certain day. X is
A. finite discrete
B. infinite discrete
C. continuous
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(s) (12 points) Determine whether each of the following is a binomial experiment.
Circle your answer.
A. An experiment consists of flipping a fair coin 22 times and observing whether
the heads is face up.
Binomial Experiment
Not a Binomial Experiment
B. An experiment consists of drawing 5 cards from a standard 52-card deck, without
replacement, and noting whether each is a spade.
Binomial Experiment
Not a Binomial Experiment
C. An experiment consists of analyzing the composition of a 4-child family in which
each child was born at a different time (no twins, triplets, etc.). .
Binomial Experiment
Not a Binomial Experiment
D. An experiment consists of drawing 6 cards one at a time with replacement and
recording the suit of each card drawn.
Binomial Experiment
Not a Binomial Experiment
E. An experiment consists of rolling a fair 6-sided die until a three is rolled.
Binomial Experiment
Not a Binomial Experiment
F. An experiment consists of casting a fair dice until a 3 lands up.
Binomial Experiment
Not a Binomial Experiment
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Part II: True/False (2 points each)
2. (16 points) Indicate whether the following statements are true or false.
A. The number of ways that 6 identical shirts and 4 identical shorts can be put up to
10!
dry on a clothes line is given by
.
6! 2!
True
False
B. A combination of r distinct objects taken from a set of size n is a selection of r of
the objects (without concern for order).
True
False
C. A game is only considered fair when the expected net winnings are at least $0.
True
False
D. The total area under a probability histogram is at most 1.
True
False
E. If a random variable is infinite, it must follow binomial distribution.
True
False
F. The normal distribution curve is not symmetric about σ.
True
False
G. Let A be the average of 200 numbers. Let B be the average of the first 100 numbers
B+C
.
and C be the average of the second 100 numbers. Then A =
2
True
False
H. If the mean for a binomial distribution is 1.2 and there were six trials, then p = 0.2.
True
False
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Part III: Work Out Problems
3. (4 points) The following is a list of the amount of rainfall in Seattle for each day of September
(measured in inches). Find the mean, median, mode, and standard deviation.
0, 0.10, 0.02, 0.01, 0.94, 0.32, 0.01, 0.02, 0, 0,
0, 0, 0, 0, 0.39, 0, 0, 0, 0, 0.16,
0.01, 0.39, 0.17, 0.07, 0.07, 0.01, 0.12, 1.11, 0.82, 0.91
4. (5 points) Taylor buys a bag of sour skittles. The bag contains 9 red skittles, 7 green skittles,
4 purple skittles, 10 orange skittles, and 5 yellow skittles. She reaches in the bag and pulls out
5 skittles at random. Red skittles are her favorite. What is the probability that at least one of
her five skittles is red?
5. The following histogram gives the probability distribution for a random variable X which takes
on the values 0, 1, 2, 3, 4, and 5.
(a) (1 point) Complete the missing rectangle
in the histogram i.e. find P (X = 3).
(b) (4 points) Find the following:
a) E(X) =
b) σ =
c) V ar(X) =
d) mode =
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6. (5 points) Stan is having a mixed stroke of luck. He just got the phone number of his waitress,
but he cannot read her handwriting. If he is certain that the first digit is a 5, the fourth digit
is a 2 or a 7, and the last digit is a 6, what is the maximum number of phone numbers Stan
must try? (Assume the phone number has only 7 digits.)
7. (5 points) Acme, Inc. ships lightbulbs in lots of 50. Before each lot is shipped, a sample of
8 lightbulbs is selected from the lot for testing. If any of the bulbs is defective, the entire lot
is rejected. What is the probability that a lot containing 3 defective lightbulbs will still get
shipped?
8. (6 points) The scores on an exam are normally distributed with a mean of 72 and a standard
deviation of 23. The instructor has decided to assign a grade of A to the top 12% of the class
and a grade of B to the next 15% of the class. What are the lowest and highest scores a student
may have on the exam and still obtain a B? Give your answer to 2 decimal places.
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9. A certain medicine is known to cause nausea in 35% of those who take it. In a group of 15
people who are taking this medicine, what is the probability that:
(a) (4 points) At least 9 will have nausea?
(b) (4 points) More than 6 but at most 10 will have nausea?
(c) (4 points) What is the expected number of people to have nausea?
10. (5 points) City Transit Authority is hiring 10 bus drivers. 20 men and 15 women apply for the
job. It is stipulated that an equal number of men and women are to be selected (5 men and 5
women). What is the probability that out of the 20 male applicants, Bob and Phil and out of
the 15 female applicants, Sara gets hired?
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11. A car insurance policy covers damages from a car accident. If you get in a major wreck, the
insurance company pays out $3000. If you get in a minor wreck, the insurance company pays
out $1000.
(a) (10 points) Suppose your monthly payment is $110. The probability that you get in a
major wreck in a given month is 0.01 and the probability you get in a minor wreck is 0.07.
Find the probability distribution for X. What is the insurance company’s expected gain?
(b) (6 points) Suppose that you provide an extra risk to the insurance company because the
probability that you get in a minor wreck jumps to 0.15. (The probability you get in a
major wreck stays the same.) What can you expect your minimum monthly payment, $a,
to be?
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12. (18 points) Bill and Sue and seven of their friends go to the movies. They all sit next to each
other in the same row. How many ways can this be done if
(a) Sue and Bill must sit next to each other?
(b) Sue must not sit next to Bill?
(c) Sue must sit in the middle seat?
(d) Sue sits on one end of the row and Bill sits on the other end of the row?
(e) Sue, Bill, or Jan sits in the middle seat?
(f) Sue, Bill, and Jan sit in the middle three seats?
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13. Mark Watney (character played by Matt Damon in The Martian) gets injured in a dust storm
and loses contact with the rest of the Ares crew while on a solo surface exploration mission on
Martian soil. He has enough food and oxygen to last him about 10 days. Each day that he
spends on the surface of Mars, he runs a 15% risk of dying solely due to radiation exposure,
8% risk of dying solely due to a suit malfunction exposing him to an average temperature of
−67◦ F and a 11% risk of dying just due to an oxygen leak. To make things worse, once he runs
out of food, each of these risk factors increase by 1% per day for every additional day that he
has to wait for his rescue team.
(a) (5 points) What is the probability of him dying due to oxygen failure or suit malfunction
within the first 5 days and solely due to excessive radiation exposure in the next 5 days?
(b) (5 points) If the crew manage to find him after 11 days of exhaustive search, then what
is the probability that that they find him still alive?
Question:
1
2
3
4
5
6
7
8
9
10
11
12
13
Total
Points:
83
16
4
5
5
5
5
6
12
5
16
18
10
190
Score:
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Rough work
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