Applied Statistics and Probability for Engineers, 7th edition 2017

CHAPTER 2

Section 2-1

Provide a reasonable description of the sample space for each of the random experiments in Exercises 2.1.1 to 2.1.11. There can be

more than one acceptable interpretation of each experiment. Describe any assumptions you make.

2.1.1. Each of four transmitted bits is classified as either in error or not in error.

2.1.2. The number of hits (views) is recorded at a high-volume Web site in a day.

2.1.3. In the final inspection of electronic power supplies, either units pass or three types of nonconformities might occur:

functional, minor, or cosmetic. Three units are inspected.

2.1.4. An ammeter that displays three digits is used to measure current in milliamperes.

2.1.5. The following two questions appear on an employee survey questionnaire. Each answer is chosen from the five-point

scale 1 (never), 2, 3, 4, 5 (always).

Is the corporation willing to listen to and fairly evaluate new ideas?

How often are my coworkers important in my overall job performance?

2.1.6. The time until a service transaction is requested of a computer to the nearest millisecond.

2.1.7. The pH reading of a water sample to the nearest tenth of a unit.

2.1.8. The voids in a ferrite slab are classified as small, medium, or large. The number of voids in each category is

measured by an optical inspection of a sample.

2-2

2.1.9. A sampled injection-molded part could have been produced either in one of two presses or in any one of the eight

cavities in each press.

2.1.10. An order for an automobile can specify either an automatic or a standard transmission, either with or without air

conditioning, and with any one of the four colors red, blue, black, or white. Describe the set of possible orders for this

experiment.

2.1.11. Calls are repeatedly placed to a busy phone line until a connection is achieved.

2.1.12. Three attempts are made to read data in a magnetic storage device before an error recovery procedure that repositions

the magnetic head is used. The error recovery procedure attempts three repositionings before an “abort’’ message is

sent to the operator. Let

Applied Statistics and Probability for Engineers, 7th edition 2017

2-3

2.1.13. Three events are shown on the Venn diagram in the following figure:

Reproduce the figure and shade the region that corresponds to each of the following events.

(a) A (b) A B (c) (A B) C (d) (B C) (e) (A B) C

Applied Statistics and Probability for Engineers, 7th edition 2017

2-4

2.1.14 In an injection-molding operation, the length and width, denoted as X and Y, respectively, of each molded part are

evaluated. Let

A denote the event of 48 < X < 52 centimeters

B denote the event of 9 < Y < 11 centimeters

Construct a Venn diagram that includes these events. Shade the areas that represent the following:

(a) A (b) A B (c) A B (d) A B

(e) If these events were mutually exclusive, how successful would this production operation be? Would the process

produce parts with X = 50 centimeters and Y = 10 centimeters?

(a)

Applied Statistics and Probability for Engineers, 7th edition 2017

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2.1.15. A digital scale that provides weights to the nearest gram is used.

(a) What is the sample space for this experiment?

Let A denote the event that a weight exceeds 11 grams, let B denote the event that a weight is less than or equal to

15 grams, and let C denote the event that a weight is greater than or equal to 8 grams and less than 12 grams.

Describe the following events.

(b) A B (c) A B (d) A (e) A B C (f) (A C) (g) A B C (h) B C (i) A (B C)

Applied Statistics and Probability for Engineers, 7th edition 2017

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2.1.16. In light-dependent photosynthesis, light quality refers to the wavelengths of light that are important. The wavelength of

a sample of photosynthetically active radiations (PAR) is measured to the nearest nanometer. The red range is 675–700

nm and the blue range is 450–500 nm. Let A denote the event that PAR occurs in the red range, and let B denote the

event that PAR occurs in the blue range. Describe the sample space and indicate each of the following events:

(a) A (b) B (c) A B (d) A B

Let w denote the wavelength. The sample space is {w | w = 0, 1, 2, …}

2.1.17. Four bits are transmitted over a digital communications channel. Each bit is either distorted or received without

distortion. Let Ai denote the event that the ith bit is distorted, i = 1....... 4.

(a) Describe the sample space for this experiment.

(b) Are the Ai’s mutually exclusive?

Describe the outcomes in each of the following events:

(c) A1 (d) A1 (e) A1 A2 A3 A4 (f) (A1 A2) (A3 A4)

Let d and o denote a distorted bit and one that is not distorted (o denotes okay), respectively.

Applied Statistics and Probability for Engineers, 7th edition 2017

2-7

2.1.18. Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks

are summarized here:

2.1.19. In control replication, cells are replicated over a period of two days. Not until mitosis is completed can freshly

synthesized DNA be replicated again. Two control mechanisms have been identified—one positive and one negative.

Suppose that a replication is observed in three cells. Let A denote the event that all cells are identified as positive, and

let B denote the event that all cells are negative. Describe the sample space graphically and display each of the

following events:

(a) A (b) B (c) A B (d) A B

Let P and N denote positive and negative, respectively.

The sample space is {PPP, PPN, PNP, NPP, PNN, NPN, NNP, NNN}.

2.1.20. Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results

from 100 samples are summarized as follows:

2.1.21. The rise time of a reactor is measured in minutes (and fractions of minutes). Let the sample space be positive,

real numbers. Define the events A and B as follows:

A = x | x 725 and B = x | x 525.

Describe each of the following events:

(a) A (b) B (c) A B (d) A B

2.1.22. The following table summarizes 204 endothermic reactions involving sodium bicarbonate.

Applied Statistics and Probability for Engineers, 7th edition 2017

2-8

Final Temperature Conditions

Heat Absorbed (cal)

Below Target

Above Target

266 K

12

40

271 K

44

16

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 below target. Determine the number of reactions in each of the following events.

(a) A B (b) A (c) A B (d) A B (e) A B

2.1.23. A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text

phrases. How many different designs are possible?

2.1.24. Consider the hospital emergency department data given below. Let A denote the event that a visit is to hospital 1, and

let B denote the event that a visit results in admittance to any hospital.

Hospital

1

2

3

4

Total

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

Determine the number of persons in each of the following events.

(a) A B (b) A (c) A B (d) A B (e) A B

2.1.25. The article “Term Efficacy of Ribavirin Plus Interferon Alfa in the Treatment of Chronic Hepatitis C,”

[Gastroenterology (1996, Vol. 111, no. 5, pp. 1307–1312)], considered the effect of two treatments and a control for

treatment of hepatitis C. The following table provides the total patients in each group and the number that showed a

complete (positive) response after 24 weeks of treatment.

Complete Response

Total

Ribavirin plus interferon alfa

16

21

Interferon alfa

6

19

Untreated controls

0

20

Applied Statistics and Probability for Engineers, 7th edition 2017

2-9

Let A denote the event that the patient was treated with ribavirin plus interferon alfa, and let B denote the event that the

response was complete. Determine the number of patients in each of the following events.

(a) A (b) A B (c) A B (d) A B

Let |A| denote the number of elements in the set A.

2.1.26. A computer system uses passwords that contain exactly eight characters, and each character is 1 of the 26 lowercase

letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let 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. Determine the number

of passwords in each of the following events.

(a) (b) A (c) A B

(d) Passwords that contain at least 1 integer

(e) Passwords that contain exactly 1 integer

Let |A| denote the number of elements in the set A.

Section 2-2

2.2.1. A sample of two printed circuit boards is selected without replacement from a batch. Describe the (ordered)

sample space for each of the following batches:

(a) The batch contains 90 boards that are not defective, 8 boards with minor defects, and 2 boards with major defects.

(b) The batch contains 90 boards that are not defective, 8 boards with minor defects, and 1 board with major defects.

Let g denote a good board, m a board with minor defects, and j a board with major defects.

2.2.2. A sample of two items is selected without replacement from a batch. Describe the (ordered) sample space for each of

the following batches:

(a) The batch contains the items {a, b, c, d}.

(b) The batch contains the items {a, b, c, d, e, f, g}.

(c) The batch contains 4 defective items and 20 good items.

(d) The batch contains 1 defective item and 20 good items.

Applied Statistics and Probability for Engineers, 7th edition 2017

2-10

2.2.3. A wireless garage door opener has a code determined by the up or down setting of 12 switches. How many

outcomes are in the sample space of possible codes?

2.2.4. In a manufacturing operation, a part is produced by machining, polishing, and painting. If there are three machine tools,

four polishing tools, and three painting tools, how many different routings (consisting of machining, followed by

polishing, and followed by painting) for a part are possible?

2.2.5. New designs for a wastewater treatment tank have proposed three possible shapes, four possible sizes, three locations

for input valves, and four locations for output valves. How many different product designs are possible?

2.2.6. A manufacturing process consists of 10 operations that can be completed in any order. How many different production

sequences are possible?

2.2.7. A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips. Assume 10 of the chips do not

conform to customer requirements.

(a) How many different samples are possible?

(b) How many samples of five contain exactly one nonconforming chip?

(c) How many samples of five contain at least one nonconforming chip?

2.2.8. In a sheet metal operation, three notches and four bends are required. If the operations can be done in any order,

2.2.9. In the laboratory analysis of samples from a chemical process, five samples from the process are analyzed daily. In

addition, a control sample is analyzed twice each day to check the calibration of the laboratory instruments.

(a) How many different sequences of process and control samples are possible each day? Assume that the five process

samples are considered identical and that the two control samples are considered identical.

(b) How many different sequences of process and control samples are possible if we consider the five process samples

to be different and the two control samples to be identical?

(c) For the same situation as part (b), how many sequences are possible if the first test of each day must be a control

sample?

Applied Statistics and Probability for Engineers, 7th edition 2017

2-11

2.2.10. In the layout of a printed circuit board for an electronic product, 12 different locations can accommodate chips.

(a) If five different types of chips are to be placed on the board, how many different layouts are possible?

(b) If the five chips that are placed on the board are of the same type, how many different layouts are possible?

2.2.11. Consider the design of a communication system.

(a) How many three-digit phone prefixes that are used to represent a particular geographic area (such as an area code)

can be created from the digits 0 through 9?

(b) As in part (a), how many three-digit phone prefixes are possible that do not start with 0 or 1, but contain 0 or 1 as

the middle digit?

(c) How many three-digit phone prefixes are possible in which no digit appears more than once in each prefix?

2.2.12. In the design of an electromechanical product, 12 components are to be stacked into a cylindrical casing in a manner

that minimizes the impact of shocks. One end of the casing is designated as the bottom and the other end is the top.

(a) If all components are different, how many different designs are possible?

(b) If seven components are identical to one another, but the others are different, how many different designs are

possible?

(c) If three components are of one type and identical to one another, and four components are of another type and

identical to one another, but the others are different, how many different designs are possible?

Applied Statistics and Probability for Engineers, 7th edition 2017

2.2.13. A bin of 50 parts contains 5 that are defective. A sample of 10 parts is selected at random, without replacement. How

many samples contain at least four defective parts?

2.2.14. Plastic parts produced by an injection-molding operation are checked for conformance to specifications. Each tool

contains 12 cavities in which parts are produced, and these parts fall into a conveyor when the press opens. An

inspector chooses 3 parts from among the 12 at random. Two cavities are affected by a temperature malfunction that

results in parts that do not conform to specifications.

(a) How many samples contain exactly 1 nonconforming part?

(b) How many samples contain at least 1 nonconforming part?

2.2.15. A hospital operating room needs to schedule three knee surgeries and two hip surgeries in a day. Suppose that an

operating room needs to handle three knee, four hip, and five shoulder surgeries.

(a) How many different sequences are possible?

(b) How many different sequences have all hip, knee, and shoulder surgeries scheduled consecutively?

(c) How many different schedules begin and end with a knee surgery?

Section 2-3

2.3.1. The sample space of a random experiment is {a, b, c, d, e} with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respectively.

Let A denote the event {a, b, c}, 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)

2-13

2.3.2. A part selected for testing is equally likely to have been produced on any one of six cutting tools.

(a) What is the sample space?

(b) What is the probability that the part is from tool 1?

(c) What is the probability that the part is from tool 3 or tool 5?

(d) What is the probability that the part is not from tool 4?

2.3.3. An injection-molded part is equally likely to be obtained from any one of the eight cavities on a mold.

(a) What is the sample space?

(b) What is the probability that a part is from cavity 1 or 2?

(c) What is the probability that a part is from neither cavity 3 nor 4?

2.3.4. A credit card contains 16 digits between 0 and 9. However, only 100 million numbers are valid. If a number is entered

randomly, what is the probability that it is a valid number?

2.3.5. In a NiCd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple

oxidation states and that is usually found in the following states:

Nickel Charge

Proportions Found

0

0.17

+2

0.35

+3

0.33

+4

0.15

(a) What is the probability that a cell has at least one of the positive nickel-charged options?

(b) What is the probability that a cell is not composed of a positive nickel charge greater than +3?

The sample space is {0, +2, +3, and +4}.

2.3.6. A message can follow different paths through servers on a network. The sender’s message can go to one of five servers

for the first step; each of them can send to five servers at the second step; each of those can send to four servers at the

third step; and then the message goes to the recipient’s server.

(a) How many paths are possible?

(b) If all paths are equally likely, what is the probability that a message passes through the first of four servers at the

third step?

Applied Statistics and Probability for Engineers, 7th edition 2017

2-14

2.3.7. Suppose your vehicle is licensed in a state that issues license plates that consist of three digits (between 0 and 9)

followed by three letters (between A and Z). If a license number is selected randomly, what is the probability that yours

is the one selected?

2.3.8. Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks

are summarized as follows:

Shock Resistance

High

Low

Scratch

High

70

9

Resistance

Low

16

5

Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch

resistance. If a disk is selected at random, determine the following probabilities:

(a) P(A) (b) P(B) (c) P(A) (d) P(A B) (e) P(A B) (f) P(A B)

2.3.9. Magnesium alkyls are used as homogenous catalysts in the production of linear low-density polyethylene (LLDPE),

which requires a finer magnesium powder to sustain a reaction. Redox reaction experiments using four different

amounts of magnesium powder are performed. Each result may or may not be further reduced in a second step using

three different magnesium powder amounts. Each of these results may or may not be further reduced in a third step

using three different amounts of magnesium powder.

(a) How many experiments are possible?

(b) If all outcomes are equally likely, what is the probability that the best result is obtained from an experiment that

uses all three steps?

(c) Does the result in part (b) change if five, six, or seven different amounts are used in the first step? Explain.

2.3.10. An article in the Journal of Database Management [“Experimental Study of a Self-Tuning Algorithm for DBMS

Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload used in the TPC-C OLTP (Transaction Processing

Performance Council’s Version C On-Line Transaction Processing) benchmark, which simulates a typical order entry

application.

Average Frequencies and Operations in TPC-C

Transaction

Frequency

Selects

Updates

New order

43

23

11

Payment

44

4.2

3

Order status

4

11.4

0

Delivery

5

130

120

Stock level

4

0

0

Applied Statistics and Probability for Engineers, 7th edition 2017

2-15

The frequency of each type of transaction (in the second column) can be used as the percentage of each type

of transaction. The average number of selects operations required for each type of transaction is shown. Let A denote

the event of transactions with an average number of selects operations of 12 or fewer. Let B denote the event of

transactions with an average number of updates operations of 12 or fewer. Calculate the following probabilities.

(a) P(A) (b) P(B) (c) P(A B) (d) P(A B) (f) P(A B)

2.3.11. Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from

100 samples are summarized as follows:

Conforms

Yes

No

1

22

8

Supplier

2

25

5

3

30

10

Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to

specifications. If a sample is selected at random, determine the following probabilities:

(a) P(A) (b) P(B) (c) P(A) (d) P(A B) (e) P(A B) (f) P(A B)

2.3.12. Consider the hospital emergency room data is given below. Let A denote the event that a visit is to hospital 4, and let B

denote the event that a visit results in LWBS (at any hospital).

Hospital

1

2

3

4

Total

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

Determine the following probabilities.

(a) P(A B) (b) P(A) (c) P(A B) (d) P(A B) (e) P(A B)

2-16

2.3.13. Use the axioms of probability to show the following: A B

(a) For any event E, P(E) = 1 − P(E). (b) P() = 0. (c) If A is contained in B, then P(A) P(B).

2.3.14. Suppose that a patient is selected randomly from the those described. The article “Term Efficacy of Ribavirin Plus

Interferon Alfa in the Treatment of Chronic Hepatitis C,” [Gastroenterology (1996, Vol. 111, no. 5, pp. 1307–1312)],

considered the effect of two treatments and a control for treatment of hepatitis C. The following table provides the total

patients in each group and the number that showed a complete (positive) response after 24 weeks of treatment.

Complete Response

Total

Ribavirin plus interferon alfa

16

21

Interferon alfa

6

19

Untreated controls

0

20

Let A denote the event that the patient is in the group treated with interferon alfa, and let B denote the event that the

patient has a complete response.

Determine the following probabilities.

(a) P(A) (b) P(B) (c) P(A B) (d) P(A B) (e) P(A B)

2.3.15. A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text

phrases. A specific design is randomly generated by the Web server when you visit the site. If you visit the site five

times, what is the probability that you will not see the same design?

2.3.16. A hospital operating room needs to schedule three knee surgeries and two hip surgeries in a day. Suppose

that an operating room needs to schedule three knee, four hip, and five shoulder surgeries. Assume that all schedules

are equally likely.

Applied Statistics and Probability for Engineers, 7th edition 2017

2-17

Determine the probability for each of the following:

(a) All hip surgeries are completed before another type of surgery.

(b) The schedule begins with a hip surgery.

(c) The first and last surgeries are hip surgeries.

2.3.17. A computer system uses passwords that contain exactly eight characters, and each character is one of

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 probability of each of the following:

(a) A (b) B

(c) A password contains at least 1 integer.

(d) A password contains exactly 2 integers.

Applied Statistics and Probability for Engineers, 7th edition 2017

2-18

Section 2-4

2.4.1. If A, B, and C are mutually exclusive events with P(A) = 0.2, P(B) = 0.3, and P(C) = 0.4, determine the following

probabilities:

(a) P(A B C) (b) P(A B C) (c) P(A B) (d) P[(A B) C] (e) P(A B C)

2.4.2. If P(A) = 0.3, P(B) =0.2, and P(A B) = 0.1, determine the following probabilities:

(a) P(A) (b) P(A B) (c) P(A B) (d) P(A B) (e) P[(A B)] (f) P(A B)

2.4.3. A manufacturer of front lights for automobiles tests lamps under a high-humidity, high-temperature environment using

intensity and useful life as the responses of interest. The following table shows the performance of 130 lamps:

Useful Life

Satisfactory

Unsatisfactory

Intensity

Satisfactory

117

3

Unsatisfactory

8

2

(a) Find the probability that a randomly selected lamp will yield unsatisfactory results under any criteria.

(b) The customers for these lamps demand 95% satisfactory results. Can the lamp manufacturer meet this demand?

2.4.4. In the article “ACL Reconstruction Using Bone-Patellar Tendon-Bone Press-Fit Fixation: 10-Year Clinical

Results” in Knee Surgery, Sports Traumatology, Arthroscopy (2005, Vol. 13, pp. 248–255), the following causes for

knee injuries were considered:

Activity

Percentage of Knee Injuries

Contact sport

46%

Noncontact sport

44%

Activity of daily living

9%

Riding motorcycle

1%

(a) What is the probability that a knee injury resulted from a sport (contact or noncontact)?

(b) What is the probability that a knee injury resulted from an activity other than a sport?

2-19

2.4.5. Consider the hospital emergency room data given below. Let A denote the event that a visit is to hospital 4, and let B

denote the event that a visit results in LWBS (at any hospital).

Hospital

1

2

3

4

Total

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

Use the addition rules to calculate the following probabilities.

(a) P(A B) (b) P(A B) (c) P(A B)

P(A) = 4329/22,252 = 0.1945, P(B) = 953/22,252 = 0.0428, P(A B) = 242/22,252 = 0.0109,

P(A B) = (984 + 3103)/22,252 = 0.1837

2.4.6. Strands of copper wire from a manufacturer are analyzed for strength and conductivity. The results from

100 strands are as follows:

Strength

High

Low

High conductivity

74

8

Low conductivity

15

3

(a) If a strand is randomly selected, what is the probability that its conductivity is high and its strength is high?

(b) If a strand is randomly selected, what is the probability that its conductivity is low or its strength is low?

(c) Consider the event that a strand has low conductivity and the event that the strand has low strength. Are these two

events mutually exclusive?

2.4.7. A computer system uses passwords that are six characters, and each character is one of the 26 letters (a–z) or

10 integers (0–9). Uppercase letters are not used. Let A denote the event that a password begins with a vowel (either a,

e, i, o, or u), and let B denote the event that a password ends with an even number (either 0, 2, 4, 6,

or 8). Suppose a hacker selects a password at random. Determine the following probabilities:

(a) P(A) (b) P(B) (c) P(A B) (d) P(A B)

2-20

2.4.8. A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text

phrases. A specific design is randomly generated by the Web server when you visit the site. Let A denote the event that

the design color is red, and let B denote the event that the font size is not the smallest one. Use the addition rules to

calculate the following probabilities.

(a) P(A B) (b) P(A B) (c) P(A B)

P(A) = 1/4 = 0.25, P(B) = 4/5 = 0.80, P(A B) = P(A)P(B) = (1/4)(4/5) = 1/5 = 0.20

2.4.9. A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase

letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Assume all passwords are equally likely. Let A and B

denote the events that consist of passwords with only letters or only integers, respectively. Determine the following

probabilities:

(a) P(A B) (b) P(A B) (c) P(Password contains exactly 1 or 2 integers)

2.4.10. Consider the three patient groups. The article “Term Efficacy of Ribavirin Plus Interferon Alfa in the Treatment of

Chronic Hepatitis C,” [Gastroenterology (1996, Vol. 111, no. 5, pp. 1307–1312)], considered the effect of two

treatments and a control for treatment of hepatitis C. 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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