CHAPTER 4. DISCRETE RANDOM VARIABLES
4.15.1 For each problem:
a. In words, define the Random Variable X.
b. List the values that X may take on.
c. Give the distribution of X. X∼
Then, answer the questions specific to each individual problem.
Exercise 4.15.7
(Solution on p. 215.)
Six different colored dice are rolled. Of interest is the number of dice that show a “1.”
d. On average, how many dice would you expect to show a “1”?
e. Find the probability that all six dice show a “1.”
f. Is it more likely that 3 or that 4 dice will show a “1”? Use numbers to justify your answer
numerically.
Exercise 4.15.8
More than 96 percent of the very largest colleges and universities (more than 15,000 to-
tal enrollments) have some online offerings.
Suppose you randomly pick 13 such insti-
tutions.
We are interested in the number that offer distance learning courses.
(Source:
http://en.wikipedia.org/wiki/Distance_education)
d. On average, how many schools would you expect to offer such courses?
e. Find the probability that at most 6 offer such courses.
f. Is it more likely that 0 or that 13 will offer such courses? Use numbers to justify your answer
numerically and answer in a complete sentence.
Exercise 4.15.9
(Solution on p. 215.)
A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet
(Vietnamese New Year) festivities this year. Based on past years, she knows that 18% of students
attend Tet festivities. We are interested in the number of students who will attend the festivities.
d. How many of the 12 students do we expect to attend the festivities?
e. Find the probability that at most 4 students will attend.
f. Find the probability that more than 2 students will attend.
Exercise 4.15.10
Suppose that about 85% of graduating students attend their graduation. A group of 22 graduating
students is randomly chosen.
d. How many are expected to attend their graduation?
e. Find the probability that 17 or 18 attend.
f. Based on numerical values, would you be surprised if all 22 attended graduation? Justify your
answer numerically.
Exercise 4.15.11
(Solution on p. 216.)
At The Fencing Center, 60% of the fencers use the foil as their main weapon. We randomly survey
25 fencers at The Fencing Center. We are interested in the numbers that do not use the foil as their
main weapon.
d. How many are expected to not use the foil as their main weapon?
e. Find the probability that six do not use the foil as their main weapon.
f. Based on numerical values, would you be surprised if all 25 did not use foil as their main
weapon? Justify your answer numerically.
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Exercise 4.15.12
Approximately 8% of students at a local high school participate in after-school sports all four
years of high school. A group of 60 seniors is randomly chosen. Of interest is the number that
participated in after-school sports all four years of high school.
d. How many seniors are expected to have participated in after-school sports all four years of high
school?
e. Based on numerical values, would you be surprised if none of the seniors participated in after-
school sports all four years of high school? Justify your answer numerically.
f. Based upon numerical values, is it more likely that 4 or that 5 of the seniors participated in
after-school sports all four years of high school? Justify your answer numerically.
Exercise 4.15.13
(Solution on p. 216.)
The chance of having an extra fortune in a fortune cookie is about 3%. Given a bag of 144 fortune
cookies, we are interested in the number of cookies with an extra fortune. Two distributions may
be used to solve this problem. Use one distribution to solve the problem.
d. How many cookies do we expect to have an extra fortune?
e. Find the probability that none of the cookies have an extra fortune.
f. Find the probability that more than 3 have an extra fortune.
g. As n increases, what happens involving the probabilities using the two distributions? Explain
in complete sentences.
Exercise 4.15.14
There are two games played for Chinese New Year and Vietnamese New Year. They are almost
identical. In the Chinese version, fair dice with numbers 1, 2, 3, 4, 5, and 6 are used, along with
a board with those numbers. In the Vietnamese version, fair dice with pictures of a gourd, fish,
rooster, crab, crayfish, and deer are used. The board has those six objects on it, also. We will play
with bets being $1. The player places a bet on a number or object. The “house” rolls three dice. If
none of the dice show the number or object that was bet, the house keeps the $1 bet. If one of the
dice shows the number or object bet (and the other two do not show it), the player gets back his
$1 bet, plus $1 profit. If two of the dice show the number or object bet (and the third die does not
show it), the player gets back his $1 bet, plus $2 profit. If all three dice show the number or object
bet, the player gets back his $1 bet, plus $3 profit.
Let X = number of matches and Y= profit per game.
d. List the values that Y may take on. Then, construct one PDF table that includes both X & Y and
their probabilities.
e. Calculate the average expected matches over the long run of playing this game for the player.
f. Calculate the average expected earnings over the long run of playing this game for the player.
g. Determine who has the advantage, the player or the house.
Exercise 4.15.15
(Solution on p. 216.)
According
to
the
South
Carolina
Department
of
Mental
Health
web
site,
for
every
200
U.S.
women,
the
average
number
who
suffer
from
anorexia
is
one
( http://www.state.sc.us/dmh/anorexia/statistics.htm16 ). Out of a randomly chosen group of
600 U.S. women:
d. How many are expected to suffer from anorexia?
e. Find the probability that no one suffers from anorexia.
16http://www.state.sc.us/dmh/anorexia/statistics.htm
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CHAPTER 4. DISCRETE RANDOM VARIABLES
f. Find the probability that more than four suffer from anorexia.
Exercise 4.15.16
The average number of children a Japanese woman has in her lifetime is 1.37. Suppose that one
Japanese woman is randomly chosen.
(
http://www.mhlw.go.jp/english/policy/children/children-childrearing/index.html
17
MHLW’s Pamphlet)
d. Find the probability that she has no children.
e. Find the probability that she has fewer children than the Japanese average.
f. Find the probability that she has more children than the Japanese average.
Exercise 4.15.17
(Solution on p. 216.)
The
average
number
of
children
a
Spanish
woman
has
in
her
life-
time
is
1.47.
Suppose
that
one
Spanish
woman
is
randomly
chosen.
( http://www.typicallyspanish.com/news/publish/article_4897.shtml 18 ).
d. Find the probability that she has no children.
e. Find the probability that she has fewer children than the Spanish average.
f. Find the probability that she has more children than the Spanish average .
Exercise 4.15.18
Fertile (female) cats produce an average of 3 litters per year. (Source: The Humane Society of
the United States). Suppose that one fertile, female cat is randomly chosen. In one year, find the
probability she produces:
d. No litters.
e. At least 2 litters.
f. Exactly 3 litters.
Exercise 4.15.19
(Solution on p. 216.)
A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership
that carries the car. She estimates the probability that any independent dealership will have the
car will be 28%. We are interested in the number of dealerships she must call.
d. On average, how many dealerships would we expect her to have to call until she finds one that
has the car?
e. Find the probability that she must call at most 4 dealerships.
f. Find the probability that she must call 3 or 4 dealerships.
Exercise 4.15.20
Suppose that the probability that an adult in America will watch the Super Bowl is 40%. Each
person is considered independent. We are interested in the number of adults in America we must
survey until we find one who will watch the Super Bowl.
d. How many adults in America do you expect to survey until you find one who will watch the
Super Bowl?
e. Find the probability that you must ask 7 people.
f. Find the probability that you must ask 3 or 4 people.
17http://www.mhlw.go.jp/english/policy/children/children-childrearing/index.html
18http://www.typicallyspanish.com/news/publish/article_4897.shtml
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Exercise 4.15.21
(Solution on p. 216.)
A group of Martial Arts students is planning on participating in an upcoming demonstration.
6 are students of Tae Kwon Do; 7 are students of Shotokan Karate. Suppose that 8 students are
randomly picked to be in the first demonstration. We are interested in the number of Shotokan
Karate students in that first demonstration. Hint: Use the Hypergeometric distribution. Look in
the Formulas section of 4: Discrete Distributions and in the Appendix Formulas.
d. How many Shotokan Karate students do we expect to be in that first demonstration?
e. Find the probability that 4 students of Shotokan Karate are picked for the first demonstration.
f. Suppose that we are interested in the Tae Kwan Do students that are picked for the first demon-
stration. Find the probability that all 6 students of Tae Kwan Do are picked for the first
demonstration.
Exercise 4.15.22
The chance of a IRS audit for a tax return with over $25,000 in income is about 2% per year. We
are interested in the expected number of audits a person with that income has in a 20 year period.
Assume each year is independent.
d. How many audits are expected in a 20 year period?
e. Find the probability that a person is not audited at all.
f. Find the probability that a person is audited more than twice.
Exercise 4.15.23
(Solution on p. 216.)
Refer to the previous problem. Suppose that 100 people with tax returns over $25,000 are ran-
domly picked. We are interested in the number of people audited in 1 year. One way to solve this
problem is by using the Binomial Distribution. Since n is large and p is small, another discrete
distribution could be used to solve the following problems. Solve the following questions (d-f)
using that distribution.
d. How many are expected to be audited?
e. Find the probability that no one was audited.
f. Find the probability that more than 2 were audited.
Exercise 4.15.24
Suppose that a technology task force is being formed to study technology awareness among in-
structors. Assume that 10 people will be randomly chosen to be on the committee from a group
of 28 volunteers, 20 who are technically proficient and 8 who are not. We are interested in the
number on the committee who are not technically proficient.
d. How many instructors do you expect on the committee who are not technically proficient?
e. Find the probability that at least 5 on the committee are not technically proficient.
f. Find the probability that at most 3 on the committee are not technically proficient.
Exercise 4.15.25
(Solution on p. 217.)
Refer back to Exercise 4.15.12. Solve this problem again, using a different, though still acceptable,
distribution.
Exercise 4.15.26
Suppose that 9 Massachusetts athletes are scheduled to appear at a charity benefit. The 9 are ran-
domly chosen from 8 volunteers from the Boston Celtics and 4 volunteers from the New England
Patriots. We are interested in the number of Patriots picked.
d. Is it more likely that there will be 2 Patriots or 3 Patriots picked?
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CHAPTER 4. DISCRETE RANDOM VARIABLES
Exercise 4.15.27
(Solution on p. 217.)
On average, Pierre, an amateur chef, drops 3 pieces of egg shell into every 2 batters of cake he
makes. Suppose that you buy one of his cakes.
d. On average, how many pieces of egg shell do you expect to be in the cake?
e. What is the probability that there will not be any pieces of egg shell in the cake?
f. Let’s say that you buy one of Pierre’s cakes each week for 6 weeks. What is the probability that
there will not be any egg shell in any of the cakes?
g. Based upon the average given for Pierre, is it possible for there to be 7 pieces of shell in the
cake? Why?
Exercise 4.15.28
It has been estimated that only about 30% of California residents have adequate earthquake sup-
plies. Suppose we are interested in the number of California residents we must survey until we
find a resident who does not have adequate earthquake supplies.
d. What is the probability that we must survey just 1 or 2 residents until we find a California
resident who does not have adequate earthquake supplies?
e. What is the probability that we must survey at least 3 California residents until we find a Cali-
fornia resident who does not have adequate earthquake supplies?
f. How many California residents do you expect to need to survey until you find a California
resident who does not have adequate earthquake supplies?
g. How many California residents do you expect to need to survey until you find a California
resident who does have adequate earthquake supplies?
Exercise 4.15.29
(Solution on p. 217.)
Refer to the above problem. Suppose you randomly survey 11 California residents. We are inter-
ested in the number who have adequate earthquake supplies.
d. What is the probability that at least 8 have adequate earthquake supplies?
e. Is it more likely that none or that all of the residents surveyed will have adequate earthquake
supplies? Why?
f. How many residents do you expect will have adequate earthquake supplies?
The next 2 questions refer to the following: In one of its Spring catalogs, L.L. Bean® advertised footwear on
29 of its 192 catalog pages.
Exercise 4.15.30
Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise
footwear. Each page may be picked at most once.
d. How many pages do you expect to advertise footwear on them?
e. Is it probable that all 20 will advertise footwear on them? Why or why not?
f. What is the probability that less than 10 will advertise footwear on them?
Exercise 4.15.31
(Solution on p. 217.)
Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise
footwear. This time, each page may be picked more than once.
d. How many pages do you expect to advertise footwear on them?
e. Is it probable that all 20 will advertise footwear on them? Why or why not?
f. What is the probability that less than 10 will advertise footwear on them?
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g. Reminder: A page may be picked more than once. We are interested in the number of pages
that we must randomly survey until we find one that has footwear advertised on it. Define
the random variable X and give its distribution.
h. What is the probability that you only need to survey at most 3 pages in order to find one that
advertises footwear on it?
i. How many pages do you expect to need to survey in order to find one that advertises footwear?
Exercise 4.15.32
Suppose that you roll a fair die until each face has appeared at least once. It does not matter in
what order the numbers appear. Find the expected number of rolls you must make until each face
has appeared at least once.
4.15.2 Try these multiple choice problems.
For the next three problems: The probability that the San Jose Sharks will win any given game is 0.3694
based on a 13 year win history of 382 wins out of 1034 games played (as of a certain date). An upcoming
monthly schedule contains 12 games.
Let X = the number of games won in that upcoming month.
Exercise 4.15.33
(Solution on p. 217.)
The expected number of wins for that upcoming month is:
A. 1.67
B. 12
C. 382
1043
D. 4.43
Exercise 4.15.34
(Solution on p. 217.)
What is the probability that the San Jose Sharks win 6 games in that upcoming month?
A. 0.1476
B. 0.2336
C. 0.7664
D. 0.8903
Exercise 4.15.35
(Solution on p. 217.)
What is the probability that the San Jose Sharks win at least 5 games in that upcoming month
A. 0.3694
B. 0.5266
C. 0.4734
D. 0.2305
For the next two questions: The average number of times per week that Mrs. Plum’s cats wake her up at
night because they want to play is 10. We are interested in the number of times her cats wake her up each
week.
Exercise 4.15.36
(Solution on p. 217.)
In words, the random variable X =
A. The number of times Mrs. Plum’s cats wake her up each week
B. The number of times Mrs. Plum’s cats wake her up each hour
C. The number of times Mrs. Plum’s cats wake her up each night
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CHAPTER 4. DISCRETE RANDOM VARIABLES
D. The number of times Mrs. Plum’s cats wake her up
Exercise 4.15.37
(Solution on p. 217.)
Find the probability that her cats will wake her up no more than 5 times next week.
A. 0.5000
B. 0.9329
C. 0.0378
D. 0.0671
Exercise 4.15.38
(Solution on p. 217.)
People visiting video rental stores often rent more than one DVD at a time. The probability
distribution for DVD rentals per customer at Video To Go is given below. There is 5 video limit
per customer at this store, so nobody ever rents more than 5 DVDs.
x
0
1
2
3
4
5
P(X=x)
0.03
0.50
0.24
?
0.07
0.04
Table 4.17
A. Describe the random variable X in words.
B. Find the probability that a customer rents three DVDs.
C. Find the probability that a customer rents at least 4 DVDs.
D. Find the probability that a customer rents at most 2 DVDs.
Another shop, Entertainment Headquarters, rents DVDs and videogames. The probability distri-
bution for DVD rentals per customer at this shop is given below. They also have a 5 DVD limit per
customer.
x
0
1
2
3
4
5
P(X=x)
0.35
0.25
0.20
0.10
0.05
0.05
Table 4.18
E. At which store is the expected number of DVDs rented per customer higher?
F. If Video to Go estimates that they will have 300 customers next week, how many DVDs do they
expect to rent next week? Answer in sentence form.
G. If Video to Go expects 300 customers next week and Entertainment HQ projects that they will
have 420 customers, for which store is the expected number of DVD rentals for next week
higher? Explain.
H. Which of the two video stores experiences more variation in the number of DVD rentals per
customer? How do you know that?
Exercise 4.15.39
(Solution on p. 217.)
A game involves selecting a card from a deck of cards and tossing a coin. The deck has 52 cards
and 12 cards are "face cards" (Jack, Queen, or King) The coin is a fair coin and is equally likely to
land on Heads or Tails
• If the card is a face card and the coin lands on Heads, you win $6
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