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.
12http://www.typicallyspanish.com/news/publish/article_4897.shtml
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CHAPTER 5. DISCRETE RANDOM VARIABLES
Exercise 5.9.21
(Solution on p. 246.)
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 5.9.22
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 5.9.23
(Solution on p. 246.)
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 5.9.24
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 5.9.25
(Solution on p. 246.)
Refer back to Exercise 4.15.12. Solve this problem again, using a different, though still acceptable,
distribution.
Exercise 5.9.26
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?
229
e. What is the probability that all of the volunteers will be from the Celtics
f. Is it more likely that more of the volunteers will be from the Patriots or from the Celtics? How
do you know?
Exercise 5.9.27
(Solution on p. 247.)
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 5.9.28
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 5.9.29
(Solution on p. 247.)
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 3 questions refer to the following: In one of its Spring catalogs, L.L. Bean® advertised footwear on
29 of its 192 catalog pages.
Exercise 5.9.30
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 5.9.31
(Solution on p. 247.)
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.
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CHAPTER 5. DISCRETE RANDOM VARIABLES
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?
g. Suppose that 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. Do you expect to survey more than 10 pages in order to find one that advertises footwear on
it? Why?
i. What is the probability that you only need to survey at most 3 pages in order to find one that
advertises footwear on it?
j. How many pages do you expect to need to survey in order to find one that advertises footwear?
Exercise 5.9.32
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.
5.9.2 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 their 13 year win history of 382 wins out of 1034 games played (as of a certain date). Their 2005
schedule for November contains 12 games. Let X= number of games won in November 2005
Exercise 5.9.33
(Solution on p. 247.)
The expected number of wins for the month of November 2005 is:
A. 1.67
B. 12
C. 382
1043
D. 4.43
Exercise 5.9.34
(Solution on p. 247.)
What is the probability that the San Jose Sharks win 6 games in November?
A. 0.1476
B. 0.2336
C. 0.7664
D. 0.8903
Exercise 5.9.35
(Solution on p. 247.)
Find the probability that the San Jose Sharks win at least 5 games in November.
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 5.9.36
(Solution on p. 247.)
In words, the random variable X =
231
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
D. The number of times Mrs. Plum’s cats wake her up
Exercise 5.9.37
(Solution on p. 247.)
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 5.9.38
(Solution on p. 247.)
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)
0.03
0.50
0.24
?
0.07
0.04
Table 5.14
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. Write your answer using proper
notation.
D. Find the probability that a customer rents at most 2 DVDs. Write your answer using proper
notation.
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)
0.35
0.25
0.20
0.10
0.05
0.05
Table 5.15
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?
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CHAPTER 5. DISCRETE RANDOM VARIABLES
Exercise 5.9.39
(Solution on p. 247.)
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
• If the card is a face card and the coin lands on Tails, you win $2
• If the card is not a face card, you lose $2, no matter what the coin shows.
A. Find the expected value for this game (expected net gain or loss).
B. Explain what your calculations indicate about your long-term average profits and losses on
this game.
C. Should you play this game to win money?
Exercise 5.9.40
(Solution on p. 248.)
You buy a lottery ticket to a lottery that costs $10 per ticket. There are only 100 tickets available
be sold in this lottery. In this lottery there is one $500 prize, 2 $100 prizes and 4 $25 prizes. Find
your expected gain or loss.
Exercise 5.9.41
(Solution on p. 248.)
A student takes a 10 question true-false quiz, but did not study and randomly guesses each an-
swer. Find the probability that the student passes the quiz with a grade of at least 70% of the
questions correct.
Exercise 5.9.42
(Solution on p. 248.)
A student takes a 32 question multiple choice exam, but did not study and randomly guesses each
answer. Each question has 3 possible choices for the answer. Find the probability that the student
guesses more than 75% of the questions correctly.
Exercise 5.9.43
(Solution on p. 248.)
Suppose that you are perfoming the probability experiment of rolling one die. Let F be the event
of rolling a "4" or a "5". You are interested in how many times you need to roll the die in order to
obtain the first “4 or 5” as the outcome.
• p = probability of success (event F occurs)
• q = probability of failure (event F does not occur)
A. Write the description of the random variable X. What are the values that X can take on? Find
the values of p and q. What is the appropriate probability distribution for X?
B. Find the probability that the first occurrence of event F (“4” or “5”) is on the first or second
trial.
C. Find the probability that more than 4 trials are needed to obtain the first “4” or “5” when rolling
the die.
**Exercises 38 - 43 contributed by Roberta Bloom
233
5.10 Review13
The next two questions refer to the following:
A recent poll concerning credit cards found that 35 percent of respondents use a credit card that gives them
a mile of air travel for every dollar they charge. Thirty percent of the respondents charge more than $2000
per month. Of those respondents who charge more than $2000, 80 percent use a credit card that gives them
a mile of air travel for every dollar they charge.
Exercise 5.10.1
(Solution on p. 249.)
What is the probability that a randomly selected respondent will spend more than $2000 AND
use a credit card that gives them a mile of air travel for every dollar they charge?
A. (0.30) (0.35)
B. (0.80) (0.35)
C. (0.80) (0.30)
D. (0.80)
Exercise 5.10.2
(Solution on p. 249.)
Based upon the above information, are using a credit card that gives a mile of air travel for each
dollar spent AND charging more than $2000 per month independent events?
A. Yes
B. No, and they are not mutually exclusive either
C. No, but they are mutually exclusive
D. Not enough information given to determine the answer
Exercise 5.10.3
(Solution on p. 249.)
A sociologist wants to know the opinions of employed adult women about government funding
for day care. She obtains a list of 520 members of a local business and professional women’s
club and mails a questionnaire to 100 of these women selected at random. 68 questionnaires are
returned. What is the population in this study?
A. All employed adult women
B. All the members of a local business and professional women’s club
C. The 100 women who received the questionnaire
D. All employed women with children
The next two questions refer to the following: An article from The San Jose Mercury News was concerned
with the racial mix of the 1500 students at Prospect High School in Saratoga, CA. The table summarizes the
results. (Male and female values are approximate.) Suppose one Prospect High School student is randomly
selected.
Ethnic Group
Gender
White
Asian
Hispanic
Black
American Indian
Male
400
168
115
35
16
Female
440
132
140
40
14
Table 5.16
13This content is available online at <http://cnx.org/content/m16832/1.11/>.
234
CHAPTER 5. DISCRETE RANDOM VARIABLES
Exercise 5.10.4
(Solution on p. 249.)
Find the probability that a student is Asian or Male.
Exercise 5.10.5
(Solution on p. 249.)
Find the probability that a student is Black given that the student is Female.
Exercise 5.10.6
(Solution on p. 249.)
A sample of pounds lost, in a certain month, by individual members of a weight reducing clinic
produced the following statistics:
• Mean = 5 lbs.
• Median = 4.5 lbs.
• Mode = 4 lbs.
• Standard deviation = 3.8 lbs.
• First quartile = 2 lbs.
• Third quartile = 8.5 lbs.
The correct statement is:
A. One fourth of the members lost exactly 2 pounds.
B. The middle fifty percent of the members lost from 2 to 8.5 lbs.
C. Most people lost 3.5 to 4.5 lbs.
D. All of the choices above are correct.
Exercise 5.10.7
(Solution on p. 249.)
What does it mean when a data set has a standard deviation equal to zero?
A. All values of the data appear with the same frequency.
B. The mean of the data is also zero.
C. All of the data have the same value.
D. There are no data to begin with.
Exercise 5.10.8
(Solution on p. 249.)
The statement that best describes the illustration below is:
Figure 5.1
A. The mean is equal to the median.
B. There is no first quartile.
C. The lowest data value is the median.
D. The median equals (Q1+Q3)
2
235
Exercise 5.10.9
(Solution on p. 249.)
According to a recent article (San Jose Mercury News) the average number of babies born with
significant hearing loss (deafness) is approximately 2 per 1000 babies in a healthy baby nursery.
The number climbs to an average of 30 per 1000 babies in an intensive care nursery.
Suppose that 1000 babies from healthy baby nurseries were randomly surveyed. Find the proba-
bility that exactly 2 babies were born deaf.
Exercise 5.10.10
(Solution on p. 249.)
A “friend” offers you the following “deal.” For a $10 fee, you may pick an envelope from a box
containing 100 seemingly identical envelopes. However, each envelope contains a coupon for a
free gift.
• 10 of the coupons are for a free gift worth $6.
• 80 of the coupons are for a free gift worth $8.
• 6 of the coupons are for a free gift worth $12.
• 4 of the coupons are for a free gift worth $40.
Based upon the financial gain or loss over the long run, should you play the game?
A. Yes, I expect to come out ahead in money.
B. No, I expect to come out behind in money.
C. It doesn’t matter. I expect to break even.
The next four questions refer to the following: Recently, a nurse commented that when a patient calls the
medical advice line claiming to have the flu, the chance that he/she truly has the flu (and not just a nasty
cold) is only about 4%. Of the next 25 patients calling in claiming to have the flu, we are interested in how
many actually have the flu.
Exercise 5.10.11
(Solution on p. 249.)
Define the Random Variable and list its possible values.
Exercise 5.10.12
(Solution on p. 249.)
State the distribution of X .
Exercise 5.10.13
(Solution on p. 249.)
Find the probability that at least 4 of the 25 patients actually have the flu.
Exercise 5.10.14
(Solution on p. 249.)
On average, for every 25 patients calling in, how many do you expect to have the flu?
The next two questions refer to the following: Different types of writing can sometimes be distinguished
by the number of letters in the words used. A student interested in this fact wants to study the number of
letters of words used by Tom Clancy in his novels. She opens a Clancy novel at random and records the
number of letters of the first 250 words on the page.
Exercise 5.10.15
(Solution on p. 249.)
What kind of data was collected?