Section 9.2(Problem 8) State hypothesis H0,H1 and draw conclusion

Section 9.3(Problem 1,9)

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Section 9.4(Problem 1 only a and b)

Section 9.5(Problem 1 part a)

Problem 8 on ch.9 supplement exercises

**Reserve Problem Chapter 9 Section 2 Problem 8**

Humans are known to have a mean gestation period of 280 days

(from last menstruation) with a standard deviation of about 9 days.

A hospital wondered whether there was any evidence that their

patients were at risk for giving birth prematurely. In a random

sample of 70 women, the average gestation time was 274.7 days.

**a.** Is the alternative hypothesis one- or two-sided?

**b.** What is the P-value of the test statistic?

**Reserve Problem Chapter 9 Section 3 Problem 1**

We have 5-year statistics of the average amount of wheat crop

(tons) harvested from 1 km2 per year, the results are as follows:

560, 525, 496, 543, 499. Test the hypothesis that the mean wheat

crop is 550 tons per 1 km2 per year (? = 0.05) and choose the correct

answer. Determine a 95% confidence interval on the mean

wheat crop. Determine whether the hypothesis that the mean

wheat crop is 550 tons per 1 km2 per year (? = 0.05) is true based

on the 95% confidence interval?

**Reserve Problem Chapter 9 Section 3 Problem 9**

Human oral normal body temperature is believed to be 98.6? F,

but there is evidence that it actually should be 98.2? F

[Mackowiak, Wasserman, Steven and Levine, JAMA (1992, Vol. 268(12), pp. 1578–1580)].

From a sample of 52 healthy

adults, the mean oral temperature was 98.285 with a standard

deviation of 0.625 degrees.

a. What are the null and alternative hypotheses?

b. Test the null hypothesis at ? = 0.05. Find t0. Is it possible to

reject H0 hypothesis at the 0.05 level of significance?

c. Construct a 95% confidence interval to answer the same

question.

**Reserve Problem Chapter 9 Section 4 Problem 1**

A group of 15 students has performed an experiment, they measured

the coefficient of thermal expansion for aluminum. The

results are as follows (10?6 K?1):22.0, 25.9, 25.6, 23.1, 22.7, 25.6,

24.9, 21.9, 26.1, 24.3, 23.5, 20.7, 21.4, 23.5, 20.4.

a. Is there strong evidence to conclude that the standard deviation

in this experiment exceeds 3? Use ? = 0.05.

b. Find the P-value for this test.

**Reserve Problem Chapter 9 Section 5 Problem 1**

A manufacturer of soap bubble liquid will test a new solution

formula. The solution will be approved, if the percent of produced

parisons, in which the content does not allow the bubbles

to inflate, does not exceed 7%. A random sample of 700 parisons

contains 50 defective parisons. Use the z-values rounded to three

decimal places to obtain the answers.

a. Formulate and test an appropriate set of hypotheses to determine

whether the solution can be approved. Use ? = 0.05.

Find the P-value.

**Reserve Supplemental Exercises Chapter 9 Problem 8**

A communication channel is being monitored by recording the

number of errors in a string of 1000 bits. Data for 20 of these

strings follow:

3 1 0 1 3 2 4 1 3 1

1 1 2 3 3 2 0 2 0 1

Consider the 20 observations collected on the number of errors in

a string of 1000 bits of a communication channel.

a. Based on the description of the random variable and these

20 observations, is a binomial distribution an appropriate

model? Perform a goodness-of-fit procedure with ? = 0.05.

Let the expected frequencies be at least three.

b. Calculate the P-value for this test.

Section 9.2(Problem 8) State hypothesis H0,H1 and draw conclusion

Section 9.3(Problem 1,9)

Section 9.4(Problem 1 only a and b)

Section 9.5(Problem 1 part a)

Problem 8 on ch.9 supplement exercises

Reserve Problem Chapter 9 Section 2 Problem 8

Humans are known to have a mean gestation period of 280 days

(from last menstruation) with a standard deviation of about 9 days.

A hospital wondered whether there was any evidence that their

patients were at risk for giving birth prematurely. In a random

sample of 70 women, the average gestation time was 274.7 days.

a. Is the alternative hypothesis one- or two-sided?

b. What is the P-value of the test statistic?

Reserve Problem Chapter 9 Section 3 Problem 1

We have 5-year statistics of the average amount of wheat crop

(tons) harvested from 1 km2 per year, the results are as follows:

560, 525, 496, 543, 499. Test the hypothesis that the mean wheat

crop is 550 tons per 1 km2 per year (α = 0.05) and choose the correct

answer. Determine a 95% confidence interval on the mean

wheat crop. Determine whether the hypothesis that the mean

wheat crop is 550 tons per 1 km2 per year (α = 0.05) is true based

on the 95% confidence interval?

Reserve Problem Chapter 9 Section 3 Problem 9

Human oral normal body temperature is believed to be 98.6∘ F,

but there is evidence that it actually should be 98.2∘ F

[Mackowiak, Wasserman, Steven and Levine, JAMA (1992, Vol. 268(12), pp. 1578–1580)].

From a sample of 52 healthy

adults, the mean oral temperature was 98.285 with a standard

deviation of 0.625 degrees.

a. What are the null and alternative hypotheses?

b. Test the null hypothesis at α = 0.05. Find t0. Is it possible to

reject H0 hypothesis at the 0.05 level of significance?

c. Construct a 95% confidence interval to answer the same

question.

Reserve Problem Chapter 9 Section 4 Problem 1

A group of 15 students has performed an experiment, they measured

the coefficient of thermal expansion for aluminum. The

results are as follows (10−6 K−1):22.0, 25.9, 25.6, 23.1, 22.7, 25.6,

24.9, 21.9, 26.1, 24.3, 23.5, 20.7, 21.4, 23.5, 20.4.

a. Is there strong evidence to conclude that the standard deviation

in this experiment exceeds 3? Use α = 0.05.

b. Find the P-value for this test.

Reserve Problem Chapter 9 Section 5 Problem 1

A manufacturer of soap bubble liquid will test a new solution

formula. The solution will be approved, if the percent of produced

parisons, in which the content does not allow the bubbles

to inflate, does not exceed 7%. A random sample of 700 parisons

contains 50 defective parisons. Use the z-values rounded to three

decimal places to obtain the answers.

a. Formulate and test an appropriate set of hypotheses to determine

whether the solution can be approved. Use α = 0.05.

Find the P-value.

Reserve Supplemental Exercises Chapter 9 Problem 8

A communication channel is being monitored by recording the

number of errors in a string of 1000 bits. Data for 20 of these

strings follow:

3101324131

1123320201

Consider the 20 observations collected on the number of errors in

a string of 1000 bits of a communication channel.

a. Based on the description of the random variable and these

20 observations, is a binomial distribution an appropriate

model? Perform a goodness-of-fit procedure with α = 0.05.

Let the expected frequencies be at least three.

b. Calculate the P-value for this test.

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Reserve Problems

The following problems have been reserved for your use in assignments and testing and do not

appear in student versions of the text. Those marked with a WileyPLUS icon have been authored

for assignment and automatic grading in WileyPLUS.

WP Problem available in WileyPLUS at instructor’s discretion.

Chapter 2 Exercises

Exercises for Section 2.1

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. WP Reserve Problem Chapter 2 Section 1 Problem 1

Determine the sample space for the following random experiment. It is known that the delivery time is within 10 to 46 hours.

Consider an experiment that records the time from an order until

the shipment arrives.

. WP Reserve Problem Chapter 2 Section 1 Problem 2

Consider an experiment that records the number of bushels (and

fractions of bushels) of corn produced in an acre. The yield is

assumed to be between 130 and 230 bushels. Determine the corresponding sample space.

. WP Reserve Problem Chapter 2 Section 1 Problem 3

Consider an experiment that records the error (diﬀerence)

between the actual and estimated quarterly revenue of a corporation. Determine the corresponding sample space.

. WP Reserve Problem Chapter 2 Section 1 Problem 4

A conceptual smartphone design uses a parachute technique to

avoid severe damage to the screen and phone-case. Each of four

nozzles located on the case might be either functional or defective

after a year.

a. Describe the sample space in terms of the condition (functional or defective) of each nozzle after a year. Let “F”

denote a functional nozzle after a year and “D” denote a

defective one.

b. How many outcomes are in the event deﬁned by two defective nozzles?

. WP Reserve Problem Chapter 2 Section 1 Problem 5

Let X denote the grams of gold obtained in a ton of ore. Consider

the two events A = {x|0 ≤ x < 3} and B = {x|2 < x < 4}. Determine
the following events.
a. A ∩ B b. A ∪ B
d. A′ ∩ B′
c. A′
. WP Reserve Problem Chapter 2 Section 1 Problem 6
Each of three machined parts is classiﬁed as either above or
below the target speciﬁcation for the part. Let a and b denote
a part above and below the speciﬁcation, respectively. Provide
a reasonable description of the sample space for this random
experiment.
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. WP Reserve Problem Chapter 2 Section 1 Problem 7
Each of 24 Web sites is classiﬁed as containing or not containing
banner ads. Provide a reasonable description of the sample space
for this random experiment.
. WP Reserve Problem Chapter 2 Section 1 Problem 8
A scale that displays two decimal places is used to measure material feeds in a chemical plant in tons. Choose a reasonable description of the sample space for this random experiment.
. WP Reserve Problem Chapter 2 Section 1 Problem 9
Provide a reasonable description of the sample space for a measurement of the concentration of ozone to the nearest part per
billion.
. WP Reserve Problem Chapter 2 Section 1 Problem 10
The time of a chemical reaction is recorded to the nearest millisecond. Provide a reasonable description of the sample space for this
experiment.
. WP Reserve Problem Chapter 2 Section 1 Problem 11
An order for a computer system can specify memory of 4, 8,
or 12 gigabytes and disk storage of 200, 300, or 400 gigabytes.
Choose the diagrams that describe the set of possible orders.
. WP Reserve Problem Chapter 2 Section 1 Problem 12
Three events are shown on the Venn diagram in the following
ﬁgure:
A
B
C
Match the ﬁgures and the corresponding events.
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Reserve Problems
. WP Reserve Problem Chapter 2 Section 1 Problem 13
Samples of a cast aluminum part are classiﬁed on the basis of
surface ﬁnish (in microinches) and edge ﬁnish. The results of 100
parts are summarized as follows:
Edge Finish
Excellent
Surface Finish
Good
Excellent
70
1
Good
12
17
Let A denote the event that a sample has excellent surface ﬁnish,
and let B denote the event that a sample has excellent edge ﬁnish. Determine the number of samples in each of the following
events.
. WP Reserve Problem Chapter 2 Section 1 Problem 14
Counts of the Web pages provided by each of two computer
servers in a selected hour of the day are recorded. Let A denote
the event that at least 10 pages are provided by server 1, and let
B denote the event that at least 20 pages are provided by server 2.
For each of the following events choose the corresponding sample
space graph:
a. A
b. B
c. A ∩ B d. A ∪ B
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. WP Reserve Problem Chapter 2 Section 1 Problem 15
A reactor’s rise time is measured in minutes (and fractions of
minutes). Let the sample space for the rise time of each batch be
positive, real numbers. Consider the rise times of two batches. Let
A denote the event that the rise time of batch 1 is less than 72.5
minutes, and let B denote the event that the rise time of batch 2 is
greater than 52.5 minutes. For each of the following events choose
the corresponding sample space graph:
a. A
b. B’
c. A ∩ B d. A ∪ B
. WP Reserve Problem Chapter 2 Section 1 Problem 16
A byte is a sequence of eight bits and each bit is either 0 or 1.
a. How many diﬀerent bytes are possible?
b. If the ﬁrst bit of a byte is a parity check, that is, the ﬁrst byte
is determined from the other seven bits, how many diﬀerent
bytes are possible?
. WP Reserve Problem Chapter 2 Section 1 Problem 17
In a chemical plant, 24 holding tanks are used for ﬁnal
product storage. Four tanks are selected at random and without
replacement. Suppose that four of the tanks contain material in
which the viscosity exceeds the customer requirements.
a. What is the probability that exactly one tank in the sample
contains high-viscosity material?
b. What is the probability that at least one tank in the sample
contains high-viscosity material?
c. In addition to the four tanks with high-viscosity levels, four
diﬀerent tanks contain material with high impurities. What is
the probability that exactly one tank in the sample contains
high-viscosity material and exactly one tank in the sample
contains material with high impurities?
. WP Reserve Problem Chapter 2 Section 1 Problem 18
An article in The Journal of Data Science [“A Statistical Analysis of Well Failures in Baltimore County” (2009, Vol. 7,
pp. 111-127)] provided the following table of well failures for
diﬀerent geological formation groups in Baltimore County.
Wells
Geological Formation Group
Failed
Total
Gneiss
170
1685
Granite
2
28
Loch raven schist
443
3733
Maﬁc
14
363
Marble
29
309
Prettyboy schist
60
1403
Other schists
46
933
Serpentine
3
39
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Let A denote the event that the geological formation has more than
1000 wells, and let B denote the event that a well failed. Determine
the number of wells in each of the following events.
a. A ∩ B
b. A′
c. A ∪ B
′
e. A′ ∩ B′
d. A ∪ B
Exercises for Section 2.2
. WP Reserve Problem Chapter 2 Section 2 Problem 1
Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of
orders in the previous month. The number of customers in each
category are shown in the following table. Customers are to be
selected without replacement.
Shipping Contract
Express
Standard
Total
No orders
25
15
40
One order
65
45
110
More than one order
40
20
60
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Reserve Problems
a. How many diﬀerent samples of eight express customers are
possible?
b. How many diﬀerent samples of ten standard customers are
possible?
c. How many diﬀerent samples of eight express and ten standard customers are possible?
. WP Reserve Problem Chapter 2 Section 2 Problem 2
Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of
orders in the previous month. The number of customers in each
category are shown in the following table. Customers are to be
selected without replacement.
Shipping Contract
k
Express
Standard
Total
No orders
25
15
40
One order
65
45
110
More than one order
40
20
60
a. Suppose that 10 express customers are selected without
replacement. How many samples contain exactly one customer with more than one order last month?
b. Suppose that 10 express customers are selected without
replacement. How many samples contain at least one customer with more than one order last month?
c. Suppose that 10 express and 15 standard customers are
selected without replacement. How many samples contain
exactly one express customer and exactly one standard customer with no orders last month?
8:50pm Page R-3
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. WP Reserve Problem Chapter 2 Section 2 Problem 3
A committee will be formed with 4 managers and 3 engineers
selected without replacement from 10 managers and 20 engineers.
How many diﬀerent committees are possible?
. WP Reserve Problem Chapter 2 Section 2 Problem 4
Code 39 is a common bar code system that consists of narrow
and wide bars (black) separated by either wide or narrow spaces
(white). Each character contains nine elements (ﬁve bars and four
spaces). The code for a character starts and ends with a bar (either
narrow or wide) and a (white) space appears between each bar.
The original speciﬁcation (since revised) used exactly two wide
bars and one wide space in each character. For example, if b and
B denote narrow and wide (black) bars, respectively, and w and W
denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). One character is held back as a start
and stop delimiter. For each of the following cases, how many
characters can be encoded?
a. The constraint of exactly two wide bars is replaced with one
that requires exactly one wide bar.
b. The constraint of exactly two wide bars is replaced with one
that allows either one or two wide bars.
c. The constraint of exactly two wide bars is dropped.
d. The constraints of exactly two wide bars and one wide space
are dropped.
k
Exercises for Section 2.3
. WP Reserve Problem Chapter 2 Section 3 Problem 1
In a car racing competition, four car brands are competing, each
having two cars representing them. Cars from the same company have the same probability of winning. Each of the cars from
Acura, Alfa Romeo and Audi has the same probability of winning
the match, while each of Aston Martin’s cars has the probability
of winning 1.57 times the probability of winning for each of the
other brands’ cars.
a. What is the chance of winning for each of the Aston Martin’s
cars? What is the chance of winning for each other cars?
b. What is the probability of winning for Aston Martin brand?
What is the probability of winning for each other brand?
. WP Reserve Problem Chapter 2 Section 3 Problem 2
Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number
of orders in the previous month. The number of customers in each
category are shown in the following table.
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Shipping Contract
Express
Standard
Total
No orders
25
15
40
One order
65
44
109
More than one order
44
20
64
Total
134
79
213
Suppose that three customers are selected randomly, without
replacement, for a survey.
a. What is the probability that at least two selected customers
use express shipping?
b. What is the probability that customers from all three order
categories are selected?
. WP Reserve Problem Chapter 2 Section 3 Problem 3
A committee will be formed with 2 managers and 4 engineers
selected randomly without replacement from 9 managers and
18 engineers.
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Reserve Problems
a. What is the probability that the speciﬁc engineer Jane and
the speciﬁc manager Mary are on the committee?
b. Each of the engineers diﬀer in years of experience. What is
the probability that the most experienced and least experienced engineers are on the committee?
. WP Reserve Problem Chapter 2 Section 3 Problem 4
Each of the possible ﬁve outcomes of a random experiment is
equally likely. The sample space is [a, b, c, d, e]. Let A denote the
event [a, b], and let B denote the event [c, d, e]. Determine the
following:
a. P(A) =
b. P(B) =
c. P(A′ ) =
d. P(A ∪ B) =
e. P(A ∩ B) =
WP
Reserve Problem Chapter 2 Section 3 Problem 5
.
The following table summarizes 186 endothermic reactions
involving sodium bicarbonate.
Final Temperature
Conditions
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Heat absorbed (cal)
Below Target
Above Target
266 K
12
32
271 K
44
16
274 K
46
36
Let A denote the event that a reaction’s ﬁnal temperature is 271
K or less. Let B denote the event that the heat absorbed is above
target. Determine the following probabilities.
a. P(A ∩ B) =
b. P(A′ ) =
c. P(A ∪ B) =
d. P(A ∪ B′ ) =
e. P(A′ ∩ B′ ) =
. WP Reserve Problem Chapter 2 Section 3 Problem 6
An article in The Journal of Data Science [“A Statistical Analysis of Well Failures in Baltimore County” (2009, Vol. 7,
pp. 111-127)] provided the following table of well failures for
diﬀerent geological formation groups in Baltimore County.
Wells
Geological Formation Group
Failed
Total
Gneiss
Granite
Loch raven schist
Maﬁc
Marble
Prettyboy schist
Other schists
Serpentine
170
2
443
14
29
60
46
3
1685
28
3733
363
309
1403
933
39
Let A denote the event that the geological formation has more than
1000 wells, and let B denote the event that a well failed. Determine
the following probabilities.
a. P(A ∩ B) =
b. P(A′ ) =
c. P(A ∪ B) =
d. P(A ∪ B′ ) =
e. P(A′ ∩ B′ ) =
. WP Reserve Problem Chapter 2 Section 3 Problem 7
Code 39 is a common bar code system that consists of narrow
and wide bars (black) separated by either wide or narrow spaces
(white). Each character contains nine elements (ﬁve bars and four
spaces). The code for a character starts and ends with a bar (either
narrow or wide) and a (white) space appears between each bar.
The original speciﬁcation (since revised) used exactly two wide
bars and one wide space in each character. For example, if b and B
denote narrow and wide (black) bars, respectively, and w and W
denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally
likely (none is held back as a delimiter). Determine the probability
for each of the following:
a. A wide space occurs before a narrow space.
b. Two wide bars occur consecutively.
c. Two consecutive wide bars are at the start or end.
d. The middle bar is wide.
Exercises for Section 2.4
. WP Reserve Problem Chapter 2 Section 4 Problem 1
Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of
orders in the previous month. The number of customers in each
category are shown in the following table. Customers are to be
selected without replacement.
Shipping Contract
Express
Standard
Total
No orders
One order
More than one order
22
64
44
17
44
21
39
108
65
a. What is the probability that a randomly selected customer
had at least one order in the previous month or uses express
shipping?
b. What is the probability that a randomly selected customer
uses standard shipping or had no orders?
. WP Reserve Problem Chapter 2 Section 4 Problem 2
A committee will be formed with 4 managers and 4 engineers
selected randomly without replacement from 11 managers and 16
engineers. What is the probability that engineer Jane or manager
Mary is on the committee?
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Reserve Problems
. WP Reserve Problem Chapter 2 Section 4 Problem 3
Consider the hospital emergency department visits shown in table
below. People may leave without being seen by a physician, and
those visits are denoted as LWBS. The remaining visits are serviced at the emergency department, and the visitor may or may
not be admitted for a stay in the hospital.
Failed
Total
Gneiss
110
1485
Granite
2
28
Loch raven schist
443
3733
2
3
4
Total
Maﬁc
14
363
Total
5337
7048
5754
4405
22544
Marble
29
309
60
1403
LWBS
190
267
247
248
952
Prettyboy schist
Admitted
1267
1538
669
964
4438
Other schists
46
933
17154
Serpentine
3
39
5243
4838
3193
Assume that the record is reviewed from a visit selected randomly
from the table.
a. What is the probability that the visit is LWBS or that the visit
is from hospital 1?
b. What is the probability that the visit is not LWBS and the
hospital is 1 or 2?
. WP Reserve Problem Chapter 2 Section 4 Problem 4
Consider the endothermic reactions in the table below.
Final Temperature
Conditions
k
Geological Formation Group
1
3880
Heat absorbed (cal)
Below Target
Above Target
266 K
21
40
271 K
44
27
274 K
56
36
Let A denote the event that a reaction’s ﬁnal temperature is
271 K or less. Let B denote the event that the heat absorbed is
above target. Use the addition rules to calculate the following
probabilities.
a. P(A ∪ B) =
b. P(A ∩ B′ ) =
c. P(A′ ∪ B′ ) =
. WP Reserve Problem Chapter 2 Section 4 Problem 5
Consider the well failure data in the table below.
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Wells
Hospital
Not admitted
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Let A denote the event that the geological formation of a well
has more than 1000 wells, and let B denote the event that a
well failed. Use the addition rules to calculate the following
probabilities.
a. P(A ∪ B) =
b. P(A ∪ B′ ) =
c. P(A′ ∪ B′ ) =
. WP Reserve Problem Chapter 2 Section 4 Problem 6
Code 39 is a common bar code system that consists of narrow
and wide bars (black) separated by either wide or narrow spaces
(white). Each character contains nine elements (ﬁve bars and four
spaces). The code for a character starts and ends with a bar (either
narrow or wide) and a (white) space appears between each bar.
The original speciﬁcation (since revised) used exactly two wide
bars and one wide space in each character. For example, if b and
B denote narrow and wide (black) bars, respectively, and w and W
denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally
likely (none is held back as a delimiter). Determine the probability
for each of the following:
a. The ﬁrst bar is wide or the second bar is wide.
b. Neither the ﬁrst nor the second bar is wide.
c. The ﬁrst bar is wide or the second bar is not wide.
d. The ﬁrst bar is wide or the ﬁrst space is wide.
Exercises for Section 2.5
. WP Reserve Problem Chapter 2 Section 5 Problem 1
Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of
orders in the previous month. The number of customers in each
category are shown in the following table. Customers are to be
selected randomly.
Shipping Contract
Express
Standard
Total
No orders
22
17
39
One order
62
44
106
More than one order
40
23
63
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a. What is the conditional probability that the customer had one
order last month given that the customer uses express shipping?
b. What is the conditional probability that the customer uses
express shipping given that the customer had at least one
order last month?
. WP Reserve Problem Chapter 2 Section 5 Problem 2
Consider the hospital emergency department visits shown in table
below. People may leave without being seen by a physician, and
those visits are denoted as LWBS. The remaining visits are serviced at the emergency department, and the visitor may or may
not be admitted for a stay in the hospital.
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Reserve Problems
Hospital
1
2
3
4
Total
Total
LWBS
Admitted
Not admitted
5295
198
1277
3820
7172
271
1578
5323
5748
241
679
4828
4445
238
1014
3193
22660
948
4548
17164
Assume that the records are reviewed from two visits selected
randomly without replacement from the table.
a. What is the probability that the second visit selected is
LWBS given that the ﬁrst visit selected is LWBS?
b. What is the probability that both visits are LWBS from
hospital 4?
. WP Reserve Problem Chapter 2 Section 5 Problem 3
A committee will be formed with 4 managers and 3 engineers
selected randomly without replacement from 11 managers and 19
engineers. What is the conditional probability that engineer Al is
on the committee given that engineer Jane is on the committee?
. WP Reserve Problem Chapter 2 Section 5 Problem 4
Consider the data on wafers classiﬁed by contamination and location in the table below.
Number of
Contamination Particles
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0
1
2
3
4
5 or more
Totals
Center
0.28
0.17
0.10
0.06
0.04
0.07
0.72
Edge
0.12
0.03
0.05
0.04
0.01
0.03
0.28
a. Suppose a hacker selects a password at random. What is the
probability that your password is selected?
b. Suppose a hacker knows that your password is in event A
and selects a password at random from this subset. What is
the probability that your password is selected?
c. Suppose a hacker knows that your password is in A and B
and selects a password at random from this subset. What is
the probability that your password is selected?
. WP Reserve Problem Chapter 2 Section 5 Problem 7
If P(A|B) = 1, must A = B? Choose a correct Venn diagram to
explain your answer.
. WP Reserve Problem Chapter 2 Section 5 Problem 8
Consider the well failure data in the table below.
Wells
Geological Formation Group
Failed
Total
Gneiss
130
1685
Granite
2
28
Loch raven schist
443
3733
Maﬁc
14
363
Totals
Marble
32
309
0.40
0.20
0.15
0.10
0.05
0.10
1.00
Prettyboy schist
60
1403
Other schists
46
933
Serpentine
3
39
Assume that one wafer is selected at random from this set. Let
A denote the event that a wafer contains four or more particles,
and let B denote the event that a wafer is from the center of the
sputtering tool. Determine the following probabilities.
a. P(A) =
b. P(A| B) =
c. P(B) =
d. P(B|A) =
e. P(A ∩ B) =
f. P(A ∪ B) =
. WP Reserve Problem Chapter 2 Section 5 Problem 5
A batch of 370 samples of rejuvenated mitochondria contains 6
that are mutated (or defective). Two are selected from the batch,
at random, without replacement.
a. What is the probability that the second one selected is defective given that the ﬁrst one was defective?
b. What is the probability that both are defective?
c. What is the probability that both are acceptable?
. WP Reserve Problem Chapter 2 Section 5 Problem 6
A computer system uses passwords that are exactly 5 characters
and each character is one of the 26 letters (a–z) or 10 integers
(0–9). You maintain a password for this computer system. Let A
denote the subset of passwords that begin with a vowel (either
a, e, i, o, or u) and let B denote the subset of passwords that end
with some predetermined number (either 0, 1, 2, 3, 4, 6, 8 or 9).
Let A denote the event that the geological formation of a well has
more than 1000 wells, let B denote the event that a well failed,
and let C denote the event that the geological formation of a well
has fewer than 500 wells.
a. What is the probability of a failure given there are more than
1,000 wells in a geological formation?
b. What is the probability of a failure given there are fewer than
500 wells in a geological formation?
. WP Reserve Problem Chapter 2 Section 5 Problem 9
Code 39 is a common bar code system that consists of narrow
and wide bars (black) separated by either wide or narrow spaces
(white). Each character contains nine elements (ﬁve bars and four
spaces). The code for a character starts and ends with a bar (either
narrow or wide) and a (white) space appears between each bar.
The original speciﬁcation (since revised) used exactly two wide
bars and one wide space in each character. For example, if b and
B denote narrow and wide (black) bars, respectively, and w and W
denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally
likely (none is held back as a delimiter). Determine the probability
for each of the following.
a. The second bar is wide given that the ﬁrst bar is wide.
b. The third bar is wide given that the ﬁrst two bars are not wide.
c. The ﬁrst bar is wide given that the last bar is wide.
. WP Reserve Problem Chapter 2 Section 5 Problem 10
During a clinical trial the eﬀect of two treatments and a control for
treatment of hepatitis C were considered. The following table provides the total patients in each group and the number that showed
a complete (positive) response after 24 weeks of treatment.
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Complete Response
Total
Ribavirin plus interferon alfa
13
24
Interferon alfa
9
16
Untreated controls
0
20
Suppose that a patient is selected at random from this set. Let
A denote the event that the patient is treated with ribavirin plus
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interferon alfa, and let B denote the event that the response is
complete. Determine the following probabilities.
a. P(B| A) =
b. P(A|B) =
c. P(A|B′ ) =
d. P(A′ |B) =
Exercises for Section 2.6
k
. WP Reserve Problem Chapter 2 Section 6 Problem 1
Accidents on highways are one of the main causes of death or
injury in developing countries and the weather conditions have an
impact on the rates of death and injury. In foggy, rainy, and sunny
conditions, 1/6, 1/10, and 1/25 of the accidents result in death,
respectively. Sunny conditions occur 62% of the time, while rainy
and foggy conditions each occur 19% of the time. What is the
probability that an accident results in a death?
. WP Reserve Problem Chapter 2 Section 6 Problem 2
A committee will be formed with 2 managers and 4 engineers
selected randomly without replacement from 10 managers and 24
engineers. What is the probability that engineer Al and engineer
Jane are on the committee?
. WP Reserve Problem Chapter 2 Section 6 Problem 3
An article in the Transportation Research Part E Journal [“Arc
Routing Problems to Restore Connectivity of a Road Network”
(2016)] considered ways of re-establishing the connectivity of
road networks after a natural disaster — earthquake. Estimates
of the probabilities of a randomly chosen road being under light
debris, moderate debris, and heavy debris conditions after diﬀerent disaster magnitudes are shown in the following table. Disaster
magnitude is equally likely to be low, moderate or high.
Disaster
Magnitude
Light
Debris
Moderate
Debris
Heavy
Debris
Total
Low
Moderate
High
Total
80
50
30
160
15
40
50
105
5
10
20
35
100
100
100
300
266 K
271 K
274 K
Heat absorbed (cal)
Below Target
Above Target
19
44
56
40
11
36
k
Hospital
Total
LWBS
Admitted
Not admitted
1
2
3
4
Total
5292
235
1277
3820
6991
230
1558
5163
5640
226
666
4728
4329
262
984
3103
22,252
953
4485
16,814
Suppose that three visits that resulted in LWBS are selected randomly (without replacement) for a follow-up interview.
a. What is the probability that all three are selected from hospital 2?
b. What is the probability that all three are from the same hospital?
. WP Reserve Problem Chapter 2 Section 6 Problem 6
Consider the well failure data in the table below.
a. What is the probability that a randomly selected road is under
heavy debris after an earthquake?
b. What is the probability that a randomly selected road is under
light or moderate debris after an earthquake?
. WP Reserve Problem Chapter 2 Section 6 Problem 4
Consider the endothermic reactions in the table below.
Final Temperature
Conditions
Let A denote the event that a reaction’s ﬁnal temperature is 271 K
or less. Let B denote the event that the heat absorbed is above
target. Determine the following probabilities.
a. P(A ∩ B) =
b. P(A ∪ B) =
c. P(A′ ∪ B′ ) =
d. Use the total probability rule to determine P(A).
. WP Reserve Problem Chapter 2 Section 6 Problem 5
The following table summarizes visits to emergency departments
at four hospitals in Arizona. People may leave without being seen
by a physician, and those visits are denoted as LWBS. The remaining visits are serviced at the emergency department, and the visitor may or may not be admitted for a stay in the hospital.
k
Wells
Geological Formation Group
Failed
Total
Gneiss
Granite
Loch raven schist
Maﬁc
Marble
Prettyboy schist
Other schists
Serpentine
170
2
383
14
29
60
46
3
2885
28
3733
363
309
1403
933
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Reserve Problems
Let A denote the event that the geological formation of a well has
more than 1000 wells, and let B denote the event that a well failed.
Determine the following probabilities.
a. P(A ∩ B) =
b. P(A ∪ B) =
c. P(A′ ∪ B′ ) =
d. Use the total probability rule to determine P(A).
. WP Reserve Problem Chapter 2 Section 6 Problem 7
Consider the well failure data in the table below.
Wells
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Geological Formation Group
Failed
Total
Gneiss
160
1685
Granite
2
28
Loch raven schist
453
3733
Maﬁc
14
363
Marble
29
309
Prettyboy schist
55
1403
Other schists
51
933
Serpentine
3
39
Suppose that two failed wells are selected randomly (without
replacement) for a follow-up review.
a. What is the probability that both are from the gneiss geological formation group?
b. What is the probability that both are from the same geological formation group?
. WP Reserve Problem Chapter 2 Section 6 Problem 8
Code 39 is a common bar code system that consists of narrow
and wide bars (black) separated by either wide or narrow spaces
(white). Each character contains nine elements (ﬁve bars and four
spaces). The code for a character starts and ends with a bar (either
narrow or wide) and a (white) space appears between each bar.
The original speciﬁcation (since revised) used exactly two wide
bars and one wide space in each character. For example, if b and
B denote narrow and wide (black) bars, respectively, and w and W
denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally
likely (none is held back as a delimiter). Determine the probability
for each of the following:
a. The code starts and ends with a wide bar.
b. Two wide bars occur consecutively.
c. Two consecutive wide bars occur at the start or end
d. The middle bar is wide.
. WP Reserve Problem Chapter 2 Section 6 Problem 9
A computer system uses passwords that contain exactly 5 characters, and each character is one of the 26 lowercase letters (a–z)
or 26 uppercase letters (A–Z) or 10 integers (0–9). Let 𝛀 denote
the set of all possible passwords, and let A and B denote the
events that consist of passwords with only letters or only integers,
respectively. Suppose that all passwords in 𝛀 are equally likely.
Determine the following probabilities.
a. P(A|B′ ) =
b. P(A′ ∩ B′ ) =
c. The probability that password contains exactly 2 integers
given that it contains at least 1 integer. P =
Exercises for Section 2.7
. WP Reserve Problem Chapter 2 Section 7 Problem 1
A cell phone user selects apps to download. Each of 10 apps is
independently selected with probability 0.21.
a. If each of the ﬁrst 9 apps are downloaded, what is the probability that the last app is downloaded?
b. What is the probability that the cell phone user downloads at
least 3 apps?
c. What is the probability that app 1 or 2 is downloaded?
. WP Reserve Problem Chapter 2 Section 7 Problem 2
An article in the Transportation Research Part E Journal [“Arc
Routing Problems to Restore Connectivity of a Road Network”
(2016)] considered ways of re-establishing the connectivity of
road networks after a natural disaster — earthquake. Estimates
of the probabilities of a randomly chosen road being under light
debris, moderate debris, and heavy debris conditions after diﬀerent disaster magnitudes are shown in the following table. Disaster
magnitude is equally likely to be low, moderate or high.
Disaster
Magnitude
Low
Light
Debris
Moderate
Debris
Heavy
Debris
Total
80
15
5
100
Moderate
50
40
10
100
High
30
50
20
100
Total
160
105
35
300
Let A and B denote the events that the earthquake magnitude is
low and the road debris is heavy. Are these events independent?
. WP Reserve Problem Chapter 2 Section 7 Problem 3
Suppose that a 9% discount is independently applied to Web
orders for clothing with probability 0.150.
a. What is the probability that the ﬁrst discount is applied to the
ﬁfth order?
b. What is the probability that at least one order in the next ﬁve
receives the discount?
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. WP Reserve Problem Chapter 2 Section 7 Problem 4
Samples of emissions from three suppliers are classiﬁed for conformance to air-quality speciﬁcations. The results from 100 samples are summarized as follows:
Supplier
k
No
1
21
9
2
25
5
3
30
10
Determine P(A).
Determine P(B).
Determine P(A ∩ B).
Wells
Let A denote the event that a sample is from supplier 1, and let B
denote the event that a sample conforms to speciﬁcations.
a. Are A and B independent events?
b. Determine P(B|A).
. WP Reserve Problem Chapter 2 Section 7 Problem 5
Six tissues are extracted from an ivy plant infested by spider mites.
The plant in infested in 22% of its area. Each tissue is chosen from
a randomly selected area on the ivy plant.
a. What is the probability that there are exact four samples
showing the signs of infestation and they are successive?
b. What is the probability that there are exact three samples
showing the signs of infestation and they are three out of four
successive?
. WP Reserve Problem Chapter 2 Section 7 Problem 6
In an acid-base titration, a base or acid is gradually added to the
other until they have completely neutralized each other. Because
acids and bases are usually colorless (as are the water and salt
produced in the neutralization reaction), pH is measured to monitor the reaction. Suppose that the equivalence point is reached
after approximately 100 mL of an NaOH solution has been added
(enough to react with all the acetic acid present) but that replicates
are equally likely to indicate from 95 to 104 mL, measured to the
nearest mL. Assume that two technicians each conduct titrations
independently.
What is the probability that both technicians obtain equivalence at 101 mL?
a. What is the probability that both technicians obtain equivalence between 98 and 102 mL (inclusive)?
b. What is the probability that the average volume at equivalence from the technicians is 100 mL?
. WP Reserve Problem Chapter 2 Section 7 Problem 7
Consider the endothermic reactions given below. Let A denote
the event that a reaction’s ﬁnal temperature is 271 K or less. Let
B denote the event that the heat absorbed is above target.
Final Temperature
Conditions
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Are A and B independent events?
. WP Reserve Problem Chapter 2 Section 7 Problem 8
Consider the well failure data given below. Let A denote the event
that the geological formation of a well has more than 1000 wells,
and let B denote the event that a well failed.
Conforms
Yes
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Above Target
266K
11
41
271K
44
16
274K
56
36
Failed
Total
Gneiss
180
1985
Granite
Loch raven schist
2
28
443
3733
Maﬁc
14
363
Marble
29
309
Prettyboy schist
60
1403
Other schists
46
933
Serpentine
3
39
Determine P(A).
Determine P(B).
Determine P(A ∩ B).
Are A and B independent events?
. WP Reserve Problem Chapter 2 Section 7 Problem 9
The code 39 is a common bar code system that consists of narrow
and wide bars (black) separated by either wide or narrow spaces
(white). Each character contains nine elements (ﬁve bars and four
spaces). The code for a character starts and ends with a bar (either
narrow or wide) and a (white) space appears between each bar.
The original speciﬁcation (since revised) used exactly two wide
bars and one wide space in each character. For example, if b and
B denote narrow and wide (black) bars, respectively, and w and W
denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally
likely (none is held back as a delimiter). Let A denote the event
that the ﬁrst bar is wide and B denote the event that the second
bar is wide. Determine the following.
Determine P(A).
Determine P(B).
Determine P(A ∩ B).
Are A and B independent events?
. WP Reserve Problem Chapter 2 Section 7 Problem 10
Consider the wafers categorized by location and contamination
levels, so that the location is in the center or on the edge, and the
contamination is low or high. Let the number of wafers with low
contamination from the center and edge locations be denoted as
nlc and nle , respectively. Similarly, let nhc and nhe denote the number of wafers with high contamination from the center and edge
Heat absorbed (cal)
Below Target
Geological Formation Group
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Reserve Problems
locations, respectively. Suppose that nlc = 12nhc and nle = 12nhe .
That is, there are 12 times as many low contamination wafers as
high ones from each location. Let A denote the event that contamination is low, and let B denote the event that the location is
center.
Are A and B independent?
Does your conclusion change if the multiplier of 12 (between
low and high contamination wafers) is changed from 12 to another
positive integer?
Exercises for Section 2.8
. WP Reserve Problem Chapter 2 Section 8 Problem 1
An article in the Transportation Research Part E Journal [“Arc
Routing Problems to Restore Connectivity of a Road Network”
(2016)] considered ways of re-establishing the connectivity of
road networks after a natural disaster — earthquake. Estimates
of the probabilities of a randomly chosen road being under light
debris, moderate debris, and heavy debris conditions after diﬀerent disaster magnitudes are shown in the following table. Disaster
magnitude is equally likely to be low, moderate or high.
Disaster
Magnitude
k
Light
Debris
Moderate
Debris
Heavy
Debris
Total
Low
80
15
5
100
Moderate
50
40
10
100
High
30
50
20
100
Total
160
105
35
300
Given that a road had heavy debris after an earthquake, what is
the conditional probability that the disaster magnitude was high?
. WP Reserve Problem Chapter 2 Section 8 Problem 2
Accidents on highways are one of the main causes of death or
injury in developing countries and the weather conditions have an
impact on the rates of death and injury. In foggy, rainy, and sunny
conditions, 1/4, 1/8, and 1/21 of the accidents result in death,
respectively. Sunny conditions occur 60% of the time, while rainy
and foggy conditions each occur 20% of the time. Given that an
accident without deaths occurred, what is the conditional probability that it was foggy at the time?
. WP Reserve Problem Chapter 2 Section 8 Problem 3
Consider the endothermic reactions given below.
Final Temperature
Conditions
Use Bayes’ theorem to calculate the probability that a reaction’s
ﬁnal temperature is 271 K or less given that the heat absorbed is
above target.
. WP Reserve Problem Chapter 2 Section 8 Problem 4
Consider the well failure data given below.
Wells
Geological Formation Group
Failed
Total
Gneiss
170
1485
Granite
2
28
443
3733
Loch raven schist
Maﬁc
14
363
Marble
29
309
Prettyboy schist
60
1403
Other schists
46
933
Serpentine
3
39
Use Bayes’ theorem to calculate the probability that a randomly
selected well is in the gneiss group given that the well has
failed.
. WP Reserve Problem Chapter 2 Section 8 Problem 5
The probabilities of poor print quality given no printer problem,
misaligned paper, high ink viscosity, or printer-head debris are 0,
0.3, 0.4, and 0.6, respectively. The probabilities of no printer problem, misaligned paper, high ink viscosity, or printer-head debris
are 0.8, 0.02, 0.08, and 0.1, respectively. Determine the probability of high ink viscosity given poor print quality. Given poor print
quality, what problem is most likely?
Heat Absorbed (cal)
Below Target
Above Target
266K
13
39
271K
44
16
274K
56
36
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Exercises for Section 2.9
. WP Reserve Problem Chapter 2 Section 9 Problem 1
Decide whether a discrete or continuous random variable is the
best model for each of the following variables:
a. The time a customer spends at a grocery store is _____.
b. The number of arrivals to a grocery store on a given day is
______.
c. The number of items each customer purchases at a grocery
store is ________.
d. The weekly revenue of a grocery store is _______.
Supplemental Exercises for Chapter 2
k
. WP Reserve Supplemental Exercises Chapter 2 Problem 1
Decide whether a discrete or continuous random variable is the
best model for each of the following variables:
a. The duration of a call at a mobile phone customer care service is _______.
b. The number of busy representatives at a mobile phone customer care service at a time t is _______.
c. The number of calls made to mobile phone customer care
service on a given day is _____.
. WP Reserve Supplemental Exercises Chapter 2 Problem 2
An article in the Transportation Research Part E Journal [“Arc
Routing Problems to Restore Connectivity of a Road Network” (2016)] considered ways of re-establishing the connectivity of road networks after a natural disaster — earthquake.
Estimates of the probabilities of a randomly chosen road being
under light debris, moderate debris, and heavy debris conditions
after diﬀerent disaster magnitudes are shown in the following
table. Disaster magnitude is equally likely to be low, moderate
or high.
Disaster
Magnitude
Low
Light
Debris
Moderate
Debris
Heavy
Debris
Total
80
15
5
100
Moderate
50
40
10
100
High
30
50
20
100
Total
160
105
35
300
What is the probability that a randomly selected road is under
heavy debris given that the earthquake is either moderate or high?
. WP Reserve Supplemental Exercises Chapter 2 Problem 3
Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of orders in the previous month. The number of customers
in each category are shown in the following table. Suppose
that four customers are selected randomly, without replacement,
for a survey.
k
Shipping Contract
Express
Standard
Total
No orders
One order
More than one order
30
73
40
18
50
28
48
123
68
a. What is the probability that three selected customers use
express shipping and the other one uses standard shipping?
b. What is the conditional probability that no selected customers ordered in the previous month given that all selected
customers use express shipping?
. WP Reserve Supplemental Exercises Chapter 2 Problem 4
A conceptual smartphone design uses a parachute technique to
avoid severe damage to the screen and phone-case. Each of four
nozzles located on the case might be either functional or defective after a year. The probability that a year-old nozzle is functional is 0.66 and assume that the nozzles fail independently.
If at least two nozzles are functional, the phone will not be
damaged in a drop. Otherwise, it will necessarily be damaged
in a drop.
a. What is the probability that a year-old phone is not damaged
in a drop?
b. What is the probability that all four nozzles of a year-old
phone are functional?
c. What is the conditional probability that all four nozzles are
functional given that a year-old phone is not damaged in a
drop?
. WP Reserve Supplemental Exercises Chapter 2 Problem 5
A cell phone user selects apps to download. Each of 5 apps is
independently selected with probability 0.2. Describe the sample
space for the app downloads and determine the number of outcomes in the sample space.
. WP Reserve Supplemental Exercises Chapter 2 Problem 6
A committee will be formed with 4 managers and 6 engineers
selected randomly without replacement from 13 managers and
20 engineers. Consider the two events that engineer Jane and
manager Mary are selected for the committee. Are these events
independent?
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Hospital
Total
LWBS
Admitted
Not admitted
1
2
3
4
Total
5250
187
1239
3824
6877
255
1513
5109
5658
249
682
4727
4301
246
940
3115
22086
937
4374
16775
Determine the following probabilities.
a. P(B|A) =
b. P(A|B) =
c. P(B′ |A) =
d. P(A ∪ B′ ) =
. WP Reserve Supplemental Exercises Chapter 2 Problem 10
A sample of three calculators is selected from a manufacturing
line, and each calculator is classiﬁed as either defective or acceptable. Let A, B, and C denote the events that the ﬁrst, second,
and third calculators, respectively, are defective. Sample space
for this experiment could be described with a tree diagram. Let
“d” denote a defective calculator and let “a” denote an acceptable
calculator.
calculator 1
d
a
calculator 2
d
a
d
a
a
d
calculator 3
d
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Reserve Problems
. WP Reserve Supplemental Exercises Chapter 2 Problem 7
Customers specify delivery dates for orders. Suppose that 14 customers each independently, randomly select delivery days over
the next year (365 days).
a. What is the probability that none select the same day?
b. What is the probability that none select the same week?
Assume exactly 52 weeks in a year.
. WP Reserve Supplemental Exercises Chapter 2 Problem 8
You remove four fuses of 10, 20, 20, and 30 amperes each, but you
do not mark the corresponding circuits. If you insert the fuses so
that each sequence is equally likely, what is the probability that
the appropriate amperage fuse is assigned to all the circuits?
. WP Reserve Supplemental Exercises Chapter 2 Problem 9
Consider the hospital emergency room data from the table. Let A
denote the event that a visit is to hospital 1 and let B denote the
event that a patient is admitted to hospital 1.
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a
d
a
d
a
. WP Reserve Supplemental Exercises Chapter 2 Problem 11
Shafts are classiﬁed in terms of the machine tool that was used
for manufacturing the shaft and conformance to surface ﬁnish and
roundness.
Tool 1
Surface Finish
Conforms
Roundness Conforms
Yes
No
197
4
4
2
Yes
No
Tool 2
Roundness Conforms
Surface Finish
Conforms
Yes
145
10
Yes
No
No
4
4
a. If a shaft is selected at random, what is the probability that
the shaft conforms to surface ﬁnish requirements or to roundness requirements or is from tool 1?
b. If a shaft is selected at random, what is the probability that
the shaft conforms to surface ﬁnish requirements or does not
conform to roundness requirements or is from tool 2?
c. If a shaft is selected at random, what is the probability that the
shaft conforms to both surface ﬁnish and roundness requirements or the shaft is from tool 2?
d. If a shaft is selected at random, what is the probability that
the shaft conforms to surface ﬁnish requirements or the shaft
is from tool 2?
. WP Reserve Supplemental Exercises Chapter 2 Problem 12
The data from 200 machined parts are summarized as follows:
Depth of Bore
Edge Condition
Coarse
Moderate
Smooth
Above Target
Below Target
15
24
48
10
21
82
a. What is the probability that a part selected has a moderate
edge condition and a below-target bore depth?
b. What is the probability that a part selected has a moderate
edge condition or a below-target bore depth?
c. What is the probability that a part selected does not have a
moderate edge condition or does not have a below-target bore
depth?
. WP Reserve Supplemental Exercises Chapter 2 Problem 13
An e-mail message can travel through one of two server routes.
The probability of transmission error in each of the servers and
the proportion of messages that travel each route are shown in the
following table. Assume that the servers are independent.
S = {ddd, add, dda, ada, dad, aad, daa, aaa}
a. Use the tree diagram to describe event A
b. Use the tree diagram to describe event B
c. Use the tree diagram to describe event A ∩ B
d. Use the tree diagram to describe event B ∪ C
Probability of Error
Percentage
of Messages Server 1 Server 2 Server 3 Server 4
Route 1
55
0.01
0.015
−
−
Route 2
45
−
−
0.02
0.003
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a. What is the probability that a message will arrive without
error?
b. If a message arrives in error, what is the probability it was
sent through route 1?
. WP Reserve Supplemental Exercises Chapter 2 Problem 14
A lot of 50 spacing washers contains 30 washers that are thicker
than the target dimension. Washers are selected from the lot at
random without replacement.
a. What is the minimum number of washers that need to be
selected so that the probability that all the washers are thinner
than the target is less than 0.10?
b. What is the minimum number of washers that need to be
selected so that the probability that 1 or more washers are
thicker than the target is at least 0.90?
. WP Reserve Supplemental Exercises Chapter 2 Problem 15
Natural red hair consists of two genes. People with red hair have
two dominant genes, two regressive genes, or one dominant and
one regressive gene. A group of 1000 people was categorized as
follows:
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Gene 2
Gene 1
Dominant
Regressive
Other
Dominant
5
25
30
Regressive
7
63
35
Other
20
15
800
Let A denote the event that a person has a dominant red hair gene,
and let B denote the event that a person has a regressive red hair
gene. If a person is selected at random from this group, compute
the following:
a. P(A) =
b. P(A ∩ B) =
c. P(A ∪ B) =
d. P(A′ ∩ B) =
e. P(A|B) =
f. Probability that the selected person has red hair.
Chapter 3 Exercises
Exercises for Section 3.1
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. WP Reserve Problem Chapter 3 Section 1 Problem 1
Let us suppose that some article described that the primary charge
transfer in a thunderstorm is due to collisions between soft hail
particles and small crystals of ice at altitudes where the temperature generally ranges between −10 ∘ C and −20 ∘ C. Data are
recorded after rounding the measured temperature in this range to
the nearest one-tenth of a degree. Determine the range (possible
values) of the random variable.
. WP Reserve Problem Chapter 3 Section 1 Problem 2
Let us suppose that some article studied the probability of death
due to burn injuries. The identiﬁed risk factors in this study
are age greater than 60 years, burn injury in more than 40% of
body-surface area, and presence of inhalation injury. It is estimated that the probability of death is 0.003, 0.03, 0.33, or 0.90, if
the injured person has zero, one, two, or three risk factors, respectively. Suppose that three people are injured in a ﬁre and treated
independently. Among these three people, two people have one
risk factor and one person has three risk factors. Let the random
variable X denote number of deaths in this ﬁre. Determine the
probability mass function of X.
. WP Reserve Problem Chapter 3 Section 1 Problem 3
Let us suppose that some article investigated the probability of
corrosion of steel reinforcement in concrete structures. It is estimated that the probability of corrosion is 0.10 under speciﬁc
values of half-cell potential and concrete resistivity. The risk of
corrosion in ﬁve independent grids of a building with these values
of half-cell potential and concrete resistivity is investigated now.
Let the random variable X denote number of grids with corrosion
in this building. Determine the probability mass function of X.
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. WP Reserve Problem Chapter 3 Section 1 Problem 4
A Web site contains 100 interconnected pages. The random variable is the number of unique pages viewed by a visitor to the
Web site.
. WP Reserve Problem Chapter 3 Section 1 Problem 5
A disk drive manufacturer sells storage devices with capacities
of one terabyte, 500 gigabytes, and 100 gigabytes with probabilities 0.5, 0.3, and 0.2, respectively. The revenues associated with
the sales in that year are estimated to be $50 million, $25 million, and $10 million, respectively. Let X denote the revenue of
storage devices during that year. Determine the probability mass
function of X.
. WP Reserve Problem Chapter 3 Section 1 Problem 6
The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows:
Final Temperature
Conditions
Number of
Reactions
266 K
48
271 K
60
274 K
92
Calculate the probability mass function of X.
. WP Reserve Problem Chapter 3 Section 1 Problem 7
The following table shows the typical depth (rounded to the nearest foot) for nonfailed wells in geological formations in Baltimore
County (The Journal of Data Science, 2009, Vol. 7, pp. 111–127).
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Geological
Formation Group
Number of
Nonfailed Wells
Nonfailed
Well Depth
Gneiss
Granite
Loch Raven Schist
Maﬁc
Marble
Prettyboy Schist
Other schists
Serpentine
Total
1,515
26
3,290
349
280
1,343
887
36
7,726
255
218
317
231
267
255
267
217
2,027
Calculate the probability mass function of depth for nonfailed
wells from the table.
. WP Reserve Problem Chapter 3 Section 1 Problem 8
Consider the following circuit. The probability that each device
functions correctly is p1 = 0.7 and p2 = 0.76. Assume that devices
fail independently.
p1
p2
Determine the probability mass function of X.
Exercises for Section 3.2
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. WP Reserve Problem Chapter 3 Section 2 Problem 1
Determine the cumulative distribution function for the random
variable in Exercise 3.1.12.
. WP Reserve Problem Chapter 3 Section 2 Problem 2
Determine the cumulative distribution function for the random
variable in Exercise 3.1.14.
. WP Reserve Problem Chapter 3 Section 2 Problem 3
Determine the cumulative distribution function for the random
variable X with the probability mass function f(x) = (4/5)(1/5).
The range of X is {0, 1, 2, …}.
. WP Reserve Problem Chapter 3 Section 2 Problem 4
The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows:
. WP Reserve Problem Chapter 3 Section 2 Problem 6
The following table shows the typical depth (rounded to the
nearest foot) for nonfailed wells in geological formations in
Baltimore County (The Journal of Data Science, 2009, Vol. 7,
pp. 111–127).
Geological
Formation Group
Number of
Nonfailed Wells
Nonfailed
Well Depth
Gneiss
1,515
255
Granite
26
218
3,290
317
349
231
Loch Raven Schist
Maﬁc
280
267
Prettyboy Schist
1,343
255
48
Other schists
887
267
271 K
60
Serpentine
274 K
92
Total
Final Temperature
Conditions
Number of
Reactions
266 K
Detemine the cumulative distribution function for X.
. WP Reserve Problem Chapter 3 Section 2 Problem 5
The distribution of the time until change (in days) of a Web site
is approximated in the following table.
Days until
Changes
Probability
1.5
0.05
3.0
0.25
4.5
0.35
5.0
0.20
7.0
0.15
Marble
36
217
7,726
2,027
Detemine the cumulative distribution function for X.
Detemine the cumulative distribution function for X.
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Exercises for Section 3.3
. WP Reserve Problem Chapter 3 Section 3 Problem 1
Calculate the mean and variance for the random variable in
Exercise 3.1.12.
. WP Reserve Problem Chapter 3 Section 3 Problem 2
Calculate the mean and variance for the random variable in
Exercise 3.1.14.
. WP Reserve Problem Chapter 3 Section 3 Problem 3
The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows:
Number of Contamination
Particles
Proportion
of Wafers
0
0.30
1
0.20
2
0.15
Final Temperature
Conditions
Number of
Reactions
3
0.10
266 K
271 K
274 K
48
60
92
4
0.05
5 or more
0.20
Calculate the mean and variance for X.
. WP Reserve Problem Chapter 3 Section 3 Problem 4
The distribution of the time until change (in days) of a Web site
is approximated in the following table.
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. WP Reserve Problem Chapter 3 Section 3 Problem 6
A visual inspection of a location on wafers from a semiconductor
manufacturing process resulted in the following table.
Days until
Changes
Probability
1.5
3.0
4.5
5.0
7.0
0.05
0.25
0.35
0.20
0.15
Assume that wafers are independent with respect to contamination particles. Wafers are selected until one with ﬁve or more
contamination particles occurs. Calculate the mean and variance
for X.
. WP Reserve Problem Chapter 3 Section 3 Problem 7
Consider the following circuit. The probability that each device
functions correctly is p1 = 0.83 and p2 = 0.75. Assume that
devices fail independently.
p1
Calculate the mean for X.
Calculate the mean and variance for X.
. WP Reserve Problem Chapter 3 Section 3 Problem 5
The following table shows the typical depth (rounded to the
nearest foot) for nonfailed wells in geological formations in
Baltimore County (The Journal of Data Science, 2009, Vol. 7,
pp. 111–127).
Geological
Formation Group
Number of
Nonfailed Wells
Nonfailed
Well Depth
Gneiss
Granite
Loch Raven Schist
Maﬁc
Marble
Prettyboy Schist
Other schists
Serpentine
Total
1,515
26
3,290
349
280
1,343
887
36
7,726
255
218
317
231
267
255
267
217
2,027
Calculate the mean and variance for X.
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Exercises for Section 3.4
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. WP Reserve Problem Chapter 3 Section 4 Problem 1
The article “The Uniform Distribution as a First Practical
Approach to New Product Inventory Management” [International
Journal of Production Economics, 2008, 114(2)] proposed a uniform distribution to model the demand of a new product before
observing the actual distribution. You will model demand forecasts with a discrete uniform distribution. According to your estimates, the minimum and the maximum levels of new product
demand are 5 and 30 units per day, respectively.
a. Determine the mean and variance of new product demand.
b. Determine the probability mass function of new product
demand.
c. How do the mean and the variance of new product demand
change, if you revise your estimate of maximum demand to
be 20 instead of 30?
. WP Reserve Problem Chapter 3 Section 4 Problem 2
The article “Statistical Method on Nonrandom Clustering with
Application to Somatic Mutations in Cancer” [BMC Bioinformatics, 2011, 11(1)] developed a statistical method to discover
mutations that lead to cancer by identifying nonrandom clusters of
amino acid mutations in protein sequences in cells. In this method,
N denotes the length of a protein sequence, and n denotes the total
number of mutations in the protein. The position of a mutation is
modeled with a discrete uniform distribution between 1 and N.
Suppose that you are a biomedical engineer working on a protein
sequence with two mutations and length N = 1000. Assume that
the locations of the mutations are independent, and can be the
same for two mutations.
a. Determine the probability that the ﬁrst mutation occurs in
the ﬁrst half of the protein sequence.
b. Determine the probability that both of the mutations occur
in the ﬁrst half of the protein sequence.
c. Determine the probability that at least one of the mutations
occurs in the ﬁrst half of the protein sequence.
. WP Reserve Problem Chapter 3 Section 4 Problem 3
Show that for a discrete uniform random variable X, if each of the
values in the range of X is multiplied by the constant c, the eﬀect
is to multiply the mean of X by c and the variance of X by c2 . That
is, show that E(cX) = cE(X) and V(cX) = c2 V(X).
. WP Reserve Problem Chapter 3 Section 4 Problem 4
The number of pages in a PDF document you create has a discrete
uniform distribution from ﬁve to nine pages (including the end
points). What are the mean and standard deviation of the number
of pages in the document?
. WP Reserve Problem Chapter 3 Section 4 Problem 5
Suppose that nine-digit Social Security numbers are assigned at
random. If you randomly select a number, what is the probability that it belongs to one of the 300 million people in the United
States?
. WP Reserve Problem Chapter 3 Section 4 Problem 6
The probability that data are entered incorrectly into a ﬁeld in
a database is 0.005. A data entry form has 28 ﬁelds, and errors
occur independently for each ﬁeld. The random variable X is the
number of ﬁelds on the form with an error. Does X have a discrete
uniform distribution? Why or why not?
. WP Reserve Problem Chapter 3 Section 4 Problem 7
Consider the hospital data in the table. Suppose a patient is
selected randomly from the collection in the table. Let X denote
the hospital number of the selected patient (either 1, 2, 3, or 4).
Does X have a discrete uniform distribution? Why or why not?
Hospital
Total
LWBS
Admitted
Not admitted
1
2
3
4
Total
5350
195
1277
3878
5617
270
1558
3789
5102
246
666
4190
4472
242
984
3246
20541
953
4485
15103
Exercises for Section 3.5
. WP Reserve Problem Chapter 3 Section 5 Problem 1
Let us suppose that some article modeled the disease progression
in sepsis (a systemic inﬂammatory response syndrome (SIRS)
together with a documented infection). Both sepsis, severe sepsis
and septic shock may be life-threatening. The researchers estimate the probability of sepsis to worsen to severe sepsis or septic
shock after three days to be 0.10. Suppose that you are physician
in an intensive care unit of a major hospital, and you diagnose
four patients with sepsis.
a. What is the probability that none of the patients with sepsis
gets worse in the next three days?
b. What is the probability that all of the patients with sepsis get
worse in the next three days?
c. What is the probability that at most two patients with sepsis
get worse in the next three days?
. WP Reserve Problem Chapter 3 Section 5 Problem 2
An article in Fire Technology, 2014 (50.3) studied the eﬀectiveness of sprinklers in ﬁre control by the number of sprinklers that
activate correctly. The researchers estimate the probability of a
sprinkler to activate correctly to be 0.7. Suppose that you are an
inspector hired to write a safety report for a large ballroom with 10
sprinklers. Assume the sprinklers activate correctly or not independently.
a. What is the probability that all of the sprinklers will operate
correctly in a ﬁre?
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b. What is the probability that at least 7 of the sprinklers will
operate correctly in a ﬁre?
c. What is the minimum number of sprinklers needed so that
the probability that at least one operates correctly is at least
0.99?
. WP Reserve Problem Chapter 3 Section 5 Problem 3
Sketch the probability mass function of a binomial distribution
with n = 10 and p = 0.01 and comment on the shape of the distribution.
a. What value of X is most likely?
b. What value of X is least likely?
. WP Reserve Problem Chapter 3 Section 5 Problem 4
A particularly long traﬃc light on your morning commute is green
20% of the time that you approach it. Assume that each morning
represents an independent trial.
a. Over 5 mornings, what is the probability that the light is
green on exactly one day?
b. Over 20 mornings, what is the probability that the light is
green on exactly four days?
c. Over 20 mornings, what is the probability that the light is
green on more than four days?
. WP Reserve Problem Chapter 3 Section 5 Problem 5
Consider the patient data in table. Suppose that four patients are
randomly selected with replacement from the total for hospital 1.
Hospital
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1
2
3
4
Total
Total
5892
6991
5640
4329
22852
LWBS
195
270
246
242
953
Admitted
1277
1558
666
984
4485
Not admitted
4420
5163
4728
3103
17414
Determine the following probabilities:
a. Exactly one is LWBS.
b. Two or more are LWBS.
c. At least one is LWBS.
. WP Reserve Problem Chapter 3 Section 5 Problem 6
Assume that a Web site changes its content according to the distribution in the table below.
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b. What is the probability that the change is made in less than
4 days in 2 or fewer of the 10 updates?
c. What is the probability that at least one change is made in
less than 4 days?
d. What is the expected number of the 10 updates that occur in
less than 4 days?
. WP Reserve Problem Chapter 3 Section 5 Problem 7
Consider the endothermic reactions in the table below.
Final Temperature
Conditions
Number of
Reactions
226 K
48
271 K
60
274 K
92
A total of 20 independent reactions are to be conducted.
a. What is the probability that exactly 12 reactions result in a
ﬁnal temperature less than 272 K?
b. What is the probability that at least 19 reactions result in a
ﬁnal temperature less than 272 K?
c. What is the probability that at least 18 reactions result in a
ﬁnal temperature less than 272 K?
d. What is the expected number of reactions that result in a ﬁnal
temperature of less than 272 K?
. WP Reserve Problem Chapter 3 Section 5 Problem 8
Consider the circuit.
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pw
a
b
pw
Suppose that the probability of a device functioning correctly is
pw = 0.75 and assume that devices fail independently. What is the
probability mass function of the number of device failures?
. WP Reserve Problem Chapter 3 Section 5 Problem 9
Consider the patient data in the table below. Suppose that
patients are randomly selected with replacement from the total for
hospital 4.
Days untill
changes
Probability
1.5
0.05
1
2
3
4
Total
3.0
0.25
Total
5299
5965
5178
5047
21489
4.5
0.35
LWBS
195
270
246
242
953
5.0
0.20
Admitted
1277
1558
666
984
4485
7.0
0.15
Not admitted
3827
4137
4266
3821
16051
Hospital
Assume that 10 changes are made independently.
a. What is the probability that the change is made in less than
4 days in 7 of the 10 updates?
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What is the smallest sample size needed so that the probability is
at least 90% that at least one patient is LWBS?
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Exercises for Section 3.6
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. WP Reserve Problem Chapter 3 Section 6 Problem 1
A robot wrestling tournament with eight participants is taking
place. The defending champion is expected to win a match with
the probability of 0.85 regardless of the opponent, and matches
outcomes are assumed to be independent.
a. The single elimination tournament requires three consecutive match wins to win the tournament. What is the probability that the defending champion wins the tournament?
b. The defending champion won the tournament again and now
accepts open challenges. What is the expected number of
matches until this robot is defeated by a challenger?
c. After the ﬁrst defeat, the robot’s joints are replaced to more
ﬂexible ones, increasing the winning probability to 0.94.
What is the probability that this robot’s ﬁrst loss is the ﬁfth
challenge?
. WP Reserve Problem Chapter 3 Section 6 Problem 2
A research team has developed a face recognition device to match
photos in a database. From laboratory tests, the recognition accuracy is 92% and trials are assumed to be independent.
a. If the research team continues to run laboratory tests, what
is the mean number of trials until failure?
b. What is the probability that the ﬁrst failure occurs on the
tenth trial?
c. To improve the recognition algorithm, a chief engineer
decides to collect 10 failures. How many trials are expected
to be needed?
. WP Reserve Problem Chapter 3 Section 6 Problem 3
A fault-tolerant system that processes transactions for a ﬁnancial
services ﬁrm uses three separate computers. If the operating computer fails, one of the two spares can be immediately switched
online. After the second computer fails, the last computer can be
immediately switched online. Assume that the probability of a
failure during any transaction is 10-9 and that the transactions can
be considered to be independent events.
a. What is the mean number of transactions before all computers have failed?
b. What is the variance of the number of transactions before all
computers have failed?
. WP Reserve Problem Chapter 3 Section 6 Problem 4
Show that the probability density function of a negative binomial
random variable equals the probability density function of a geometric random variable when r = 1. Show that the formulas for
the mean and variance of a negative binomial random variable
equal the corresponding results for a geometric random variable
when r = 1.
. WP Reserve Problem Chapter 3 Section 6 Problem 5
Consider the endothermic reactions in the table below. Assume
that independent reactions are conducted.
Final Temperature
Conditions
Number of
Reactions
226 K
271 K
274 K
48
60
92
a. What is the probability that the ﬁrst reaction to result in a
ﬁnal temperature less than 272 K is the 12th reaction?
b. What is the mean reaction number of the ﬁrst ﬁnal temperature is less than 272 K?
c. What is the probability that the ﬁrst reaction to result in a
ﬁnal temperature less than 272 K occurs within 3 or fewer
reactions?
d. What is the mean number of reactions until two reactions
result in ﬁnal temperatures less than 272 K?
. WP Reserve Problem Chapter 3 Section 6 Problem 6
The following table summarizes visits to emergency departments
at four hospitals. People may leave without being seen by a physician, and those visits are denoted as LWBS. The remaining visits
are serviced at the emergency department, and the visitor may or
may not be admitted for a stay in the hospital.
Hospital
Total
LWBS
Admitted
Not admitted
1
2
3
4
Total
5463
195
1277
3991
5642
270
1558
3814
5235
246
666
4323
4029
242
984
2803
20369
953
4485
14931
Suppose that patients are randomly selected with replacement,
from the total for hospital 4. Determine the following:
a. Probability that the ﬁrst patient admitted is the ﬁrst one
selected.
b. Probability that four or fewer patients are selected to
admit two.
c. Expected number of patients selected to admit 20.
Exercises for Section 3.7
. WP Reserve Problem Chapter 3 Section 7 Problem 1
A technology company is forming a task force of six members
to deal with urgent quality issues. The positions will be ﬁlled
by randomly chosen qualiﬁed applicants. The qualiﬁed applicants
consist of ﬁve managers and ten engineers.
a. What is the probability that the chosen applicants are either
all managers or all engineers?
b. What is the probability that number of managers is the same
as the number of engineers on the task force?
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c. What is the expected number of engineers chosen?
d. What is the probability that at least one manager is chosen
for the task force?
. WP Reserve Problem Chapter 3 Section 7 Problem 2
The MIT-BIT arrhythmia database contains 48 heart signal
recordings with 22 and 26 from female and male patients, respectively. For a classiﬁcation task, an analyst randomly selects
36 records for predictive model training and keeps the other 12
records for testing the model performance.
a. What is the expected number of female patient recordings
selected for training?
b. What is the probability that at least three male patient recordings are selected for training?
c. What is the probability that the same number of male and
female patients are used for testing the model performance?
. WP Reserve Problem Chapter 3 Section 7 Problem 3
Calculate the ﬁnite population corrections for the following situations.
a. Suppose that X has a hypergeometric distribution with N =
140, n = 4, K = 50. If ﬁnite population correction factor is
small a binomial distribution can eﬀectively approximate the
hypergeometric distribution. Calculate the following probabilities, assuming that X has a binomial distribution.
b. Printed circuit cards are placed in a functional test after
being populated with semiconductor chips. A lot contains
150 cards, and 20 are selected without replacement for
functional testing. Use the binomial approximation to the
hypergeometric distribution to approximate the following
probabilities. If 23 cards are defective, what is the probability that at least 1 defective card is in the sample? If 7 cards
are defective, what is the probability that at least 1 defective
card appears in the sample?
. WP Reserve Problem Chapter 3 Section 7 Problem 4
Consider the visits that result in leave without being seen (LWBS)
at an emergency department in the table below. Assume that four
visits that result in LWBS are to be randomly selected (without
replacement) for a follow-up interview.
Hospital #
Total
1
2
3
4
Total
5292
6991
5640
4329
22,252
LWBS
195
270
246
242
953
Admitted
1277
1558
666
984
4485
Not admitted
3820
5163
4728
3103
16,814
a. What is the probability that all selected visits are from
hospital 4?
b. What is the probability that no selected visits are from
hospital 4?
c. What is the probability that all selected visits are from the
same hospital?
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. WP Reserve Problem Chapter 3 Section 7 Problem 5
Consider the nonfailed wells in the table below. Assume that
four wells are selected randomly (without replacement) for
inspection.
Geological Formation
Group
Number of
Nonfailed Wells
Nonfailed
Well Depth
1,515
26
3,290
349
280
1,343
887
36
7,726
255
218
317
231
267
255
267
217
2,027
Gneiss
Granite
Loch Raven Schist
Maﬁc
Marble
Prettyboy Schist
Other Schists
Serpentine
Total
a. What is the probability that exactly two are selected from the
Loch Raven Schist?
b. What is the probability that one or more is selected from the
Loch Raven Schist?
c. What is the expected number selected from the Loch Raven
Schist?
. WP Reserve Problem Chapter 3 Section 7 Problem 6
Consider the semiconductor wafer data in the table below.
Location in Sputtering Tool
Contamination
Low
Center
Edge
Total
514
68
582
358
High
112
246
Total
626
314
Suppose that 10 wafers are selected randomly (without replacement) for an electrical test. Determine the following:
a. Probability that exactly 4 wafers have high contamination.
b. Probability that at least 1 is from the center of the sputtering
tool and has high contamination.
c. Probability that exactly 3 have high contamination or are
from the edge of the sputtering tool.
d. Instead of 10 wafers, what is the minimum number of wafers
that need to be selected so that the probability that at least 1
wafer has high contamination is greater than or equal to 0.9?
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Exercises for Section 3.8
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. WP Reserve Problem Chapter 3 Section 8 Problem 1
An engineer is participating in a research project on the title patterns of junk emails. The number of junk emails which arrive in
an individual’s account every hour follows a Poisson distribution
with a mean of 1.2.
a. What is the expected number of junk emails that an individual receives in an eight-hour day?
b. What is the probability that an individual receives more than
two junk emails for the next three hours?
c. What is the probability that an individual receives no junk
email for two hours?
. WP Reserve Problem Chapter 3 Section 8 Problem 2
The arrivals of patients at a walk-in clinic between 1:00 and 2:00
PM follows a Poisson process with the mean of 10.5.
a. What is the probability that 15 patients arrive at the clinic
between 1:00 and 2:00 PM?
b. What is the probability that no patients arrive between 1:00
and 1:10 pm?
c. Suppose that 20 patients arrive between 1:00 and 1:30
PM. What is the probability that three more patients arrive
between 1:30 and 2:00 PM?
d. Suppose that the arrivals from 12:00 to 1:00 PM follows a
Poisson process with a mean of 4.7 Is the number of arrivals
between 12:00 and 2:00 PM a Poisson random variable?
Why or why not?
. WP Reserve Problem Chapter 3 Section 8 Problem 3
The number of ﬂaws in bolts of cloth in textile manufacturing is
assumed to be Poisson distributed with a mean of 0.1 ﬂaw per
square meter.
a. What is the probability that there are two ﬂaws in one square
meter of cloth?
b. What is the probability that there is one ﬂaw in 10 square
meters of cloth?
c. What is the probability that there are no ﬂaws in 20 square
meters of cloth?
d. What is the probability that there are at least two ﬂaws in
10 square meters of cloth?
. WP Reserve Problem Chapter 3 Section 8 Problem 4
The number of content changes to a Web site follows a Poisson
distribution with a mean of 0.25 per day.
a. What is the probability of two or more changes in a day?
b. What is the probability of no content changes in ﬁve days?
c. What is the probability of two or fewer changes in ﬁve days?
. WP Reserve Problem Chapter 3 Section 8 Problem 5
Orders arrive at a Web site according to a Poisson process with a
mean of 12 per hour. Determine the following:
a. Probability of no orders in ﬁve minutes.
b. Probability of 3 or more orders in ﬁve minutes.
c. Length of a time interval such that the probability of no
orders in an interval of this length is 0.001.
Supplemental Exercises for Chapter 3
. WP Reserve Supplemental Exercises Chapter 3 Problem 1
For each of the following exercises, determine the range (possible
values) of the random variable.
a. A store manager is interested in the customer traﬃc to a
Web site from midnight to 8:00 AM. She counts the number
unique visitors to the site in that time period.
b. An article in the Journal of Materials Science: Materials in
Medicine 2010 21(4) studies the eﬀect of the surface ﬁnish of
the metal stem on the longevity of the implants in total hip
replacement surgeries. Random damage events were examined with emitted acoustic signals under the application of
a direct shear force. The total number of random damage
events is 2624 ±235 at the yield stage. A material science
researcher models the total number of damage events with
the grit-blasted surface ﬁnish at the yield stage as a random
variable assuming that its range is within the observed values
in this article.
. WP Reserve Supplemental Exercises Chapter 3 Problem 2
Typically, a job shop is a manufacturing system in which a variety of custom products are produced in small batches. In the
study “Analysis of Reactive Scheduling Problems in a Job Shop
Environment” [European Journal of Operational Research, 2000,
126(3)], the researchers tested the performance of a variety of
job scheduling policies in a job shop system with machine breakdowns. In these tests, it is assumed that the number of operations
required to complete the production of a job has a discrete uniform distribution between 5 and 15. In addition, the processing
time of each operation has discrete uniform distribution between
20 and 80 minutes.
a. Determine the mean and variance of the number of operations required to complete the production of a job.
b. Determine the mean and variance of the processing time of
an operation.
c. Can one conclude that the mean of the total time to complete all required operations of a job is 500 minutes? Why or
why not?
. WP Reserve Supplemental Exercises Chapter 3 Problem 3
The proton exchange membrane (PEM) fuel cells produce electricity with hydrogen fuel and oxygen of the air. An article in
the Journal of Power Sources, 2008, 178(1) studied a two-phase
ﬂow (a ﬂow of gas and liquid) model applied to the gas diﬀusion
layer of PEM fuel cells. Five diﬀerent values of channel width are
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Reserve Problems
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considered with equal probability. These values are 0.3, 0.4, 0.5,
0.6 and 0.7 mm. Let the random variable X denote the channel
width. Determine the probability mass function of X.
. WP Reserve Supplemental Exercises Chapter 3 Problem 4
An article in Soil Dynamics and Earthquake Engineering 2008,
28.10 evaluated the vulnerability of buildings in Barcelona, Spain
to earthquakes. For instance, they estimate that the probability
of a high-rise, unreinforced masonry building located in zone II
of Barcelona has a moderate or severe damage is approximately
0.46. Assume that a dozen high-rise, unreinforced masonry buildings located in zone II of Barcelona respond independently.
Determine the probabilities of the following outcomes from an
earthquake:
a. All 12 buildings have moderate or severe damage
b. At least ten of these buildings have moderate or severe
damage
c. Fewer than two or more than ten of these buildings have
moderate or severe damage
. WP Reserve Supplemental Exercises Chapter 3 Problem 5
An article in The BMJ 2014, 4.9 studied the cost-eﬀectiveness
of nalmefene (a drug used mainly in the treatment of alcohol
dependence) with psychosocial support in decreasing alcohol
consumption. Suppose that the probability of a patient to remain
in abstinence (successfully respond) after a year of treatment with
nalmefene and psychosocial support is 0.20. The same probability
is estimated to be 0.08 if the patient is treated with only psychosocial support. Suppose that you are a physician who have ten new
alcohol dependent patients, and you plan to provide psychosocial
support to all of them and also provide treatment with nalmefene
to ﬁve. Assume that patient respond independently.
a. Is the number of patients who successfully respond a binomial random variable? Why or why not?
b. What is the probability that at least two patients will remain
in abstinence after a year of treatment with nalmefene and
psychosocial support?
c. What is the probability that exactly two of the ten patients
successfully respond?
(Hint: consider each group of patients separately.)
. WP Reserve Supplemental Exercises Chapter 3 Problem 6
Customers of a hardware shop make a payment either in cash or
with credit/debit card with probabilities 0.3 and 0.7, respectively.
Assume these probabilities apply to all customers independently.
a. If 20 customers pay at the hardware shop, what is the probability that exactly 5 customers pay in cash?
b. What is the probability that the second customer to pay cash
is the ﬁfth customer to pay at the shop?
c. Assume that customers arrive according to a Poisson process with a mean of 10.5 per hour. What is the probability
that there are no cash transactions in a period of 30 minutes?
. WP Reserve Supplemental Exercises Chapter 3 Problem 7
Equipment maintenance on a manufacturing line either replaces
or cleans a ﬁlter. In 85% of the maintenance events the ﬁlter is
cleaned. Assume that ﬁlter events occur independently.
a. What is the probability that the ﬁlter is ﬁrst replaced after it
is cleaned twice?
b. What is the mean and variance of the number of maintenance
events until the ﬁlter is replaced?
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c. Suppose that the number of days until maintenance events
follows a geometric distribution with a mean of 5 days. How
would you determine the mean number of days until the ﬁlter
is replaced and what assumptions would you make?
. WP Reserve Supplemental Exercises Chapter 3 Problem 8
A food company is conducting research on customers’ taste. In
each round of a blind taste experiment four black teas and one
herbal tea are presented to participants. Suppose that the participants randomly guess, independently in each round.
a. If a participant picks the herbal tea in each of three rounds, a
box of tea is won. What is the probability that a participant
wins a box of tea?
b. What is the expected number of participants until a second
box of tea is won?
. WP Reserve Supplemental Exercises Chapter 3 Problem 9
The number of contamination particles (that exceed a speciﬁed
size) in display panels for televisions follows the Poisson distribution with the mean of 0.0001 per square inch. A quality engineer
is inspecting a television panel that is 55 by 35 inches.
a. What is the probability that there is particle in the panel?
b. If there are at most two particles, the panel is sent to the
assembly shop. What is the probability that a panel is sent to
the assembly shop?
c. What is the probability that at least 9 of the next 10 panels are sent to the assembly shop? Assume that panels are
independent with respect to contamination.
. WP Reserve Supplemental Exercises Chapter 3 Problem 10
Batches that consist of 50 coil springs from a production process
are checked for conformance to customer requirements. The mean
number of nonconforming coil springs in a batch is ﬁve. Assume
that the number of nonconforming springs in a batch, denoted as
X, is a binomial random variable.
a. What are n and p?
b. What is P(X _ 2)?
c. What is P(X _ 49)?
. WP Reserve Supplemental Exercises Chapter 3 Problem 11
The probability is 0.6 that a calibration of a transducer in an electronic instrument conforms to speciﬁcations for the measurement
system. Assume that the calibration attempts are independent.
What is the probability that at most three calibration attempts are
required to meet the speciﬁcations for the measurement system?
. WP Reserve Supplemental Exercises Chapter 3 Problem 12
The probability that your call to a service line is answered in less
than 30 seconds is 0.85. Assume that your calls are independent.
a. If you call 15 times, what is the probability that exactly 9 of
your calls are answered within 30 seconds?
b. If you call 20 times, what is the probability that at least 16
calls are answered in less than 30 seconds?
c. If you call 25 times, what is the mean number of calls that
are answered in less than 30 seconds?
d. What is the probability that you must call four times to obtain
the ﬁrst answer in less than 30 seconds?
e. What is the mean number of calls until you are answered in
less than 30 seconds?
. WP Reserve Supplemental Exercises Chapter 3 Problem 13
The probability that your call to a service line is answered in less
than 30 seconds is 0.6. Assume that your calls are independent.
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Reserve Problems
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a. What is the probability that you must call six times in order
for two of your calls to be answered in less than 30 seconds?
b. What is the mean number of calls to obtain two answers in
less than 30 seconds?
. WP Reserve Supplemental Exercises Chapter 3 Problem 14
Consider the circuit shown in the ﬁgure above where the probability that each device functions correctly is provided. Assume
that all devices fail independently. What is the probability of two
or fewer failed devices?
0.9
. WP Reserve Supplemental Exercises Chapter 3 Problem 15
Determine the constant so that the following function is a probability mass function: f(x) = cx for x = 1, 2, 3, 4.
. WP Reserve Supplemental Exercises Chapter 3 Problem 16
Suppose that 50 sites on a patient might contain lesions. A biopsy
selects 8 sites randomly (without replacement).
a. What is the minimum number of sites with lesions so that
the probability of at least one selected site contains lesions
is greater than or equal to 0.95?
b. Rework the above section if the probability of that event is
greater than or equal to 0.99.
0.95
a
0.9
0.99
b
0.95
0.9
Chapter 4 Exercises
Exercises for Section 4.1
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. WP Reserve Problem Chapter 4 Section 1 Problem 1
A parabolic satellite dish reﬂects signals to the dish’s focal point.
An antenna designer analyzed signals transmitted to a satellite
( dish and) obtained the probability density function f (x) =
1
c 1 − x2 for 0 < x < 2, where X is the distance (in meters)
16
from the centroid of the dish surface to a reﬂection point at which
a signal arrives. Determine the following:
a. Value or c that makes f(x) a valid probability density function
b. P(X ≤ 0.4) =
c. P(0.1 < X < 0.4) =
. WP Reserve Problem Chapter 4 Section 1 Problem 2
The talk time (in hours) on a cell phone in a month is approxix − 10
for 10 <
mated by the probability density function f (x) =
5h
x − 25
1
for 20 ≤ x ≤ 25. Determine
x 0). Use the cumulative distribution function to
1000
Exercises for Section 4.3
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. WP Reserve Problem Chapter 4 Section 3 Problem 1
A parabolic satellite dish reﬂects signals to the dish’s focal point.
An antenna designer analyzed signals transmitted to a satellite
( dish and) obtained the probability density function f (x) =
1
c 1 − x2 for 0 < x < 3, where x is the distance (in meters)
16
from the centroid of the dish surface to a reﬂection point at which
a signal arrives. Calculate the mean and variance.
E(X) =
V(X) =
. WP Reserve Problem Chapter 4 Section 3 Problem 2
The talk time (in hours) on a cell phone in a month is approxix − 10
for 10 <
mated by the probability density function f (x) =
5h
x − 25
1
for 20 ≤ x ≤ 25. Determine
x 2.0) =
e. P(0 < Z < 0.7) =
. WP Reserve Problem Chapter 4 Section 5 Problem 4
Assume that X is normally distributed with a mean of 7 and a standard deviation of 4. Determine the value for x that solves each of
the following equations.
a. P(X > x) = 0.5

b. P(X > x) = 0.95

c. P(x < X < 9) = 0.2
d. P(3 < X < x) = 0.95
e. P(−x < X − 7 < x) = 0.99
. WP...

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