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(c) If a product does not attain a good review, what is the probability that it will be a highly successful product?
Let G denote a product that received a good review. Let H, M, and P denote products that were high, moderate, and
poor performers, respectively.
2.8.2. Suppose that P(A | B) = 0.7, P(A) 0.5, and P(B) = 0.2. Determine P(B | A).
( ) 0.5
2.8.3. A new analytical method to detect pollutants in water is being tested. This new method of chemical analysis is
important because, if adopted, it could be used to detect three different contaminants—organic pollutants, volatile
solvents, and chlorinated compounds—instead of having to use a single test for each pollutant. The makers of the test
claim that it can detect high levels of organic pollutants with 99.7% accuracy, volatile solvents with 99.95% accuracy,
and chlorinated compounds with 89.7% accuracy. If a pollutant is not present, the test does not signal. Samples are
prepared for the calibration of the test, and 60% of them are contaminated with organic pollutants, 27% with volatile
solvents, and 13% with traces of chlorinated compounds. A test sample is selected randomly.
(a) What is the probability that the test will signal?
(b) If the test signals, what is the probability that chlorinated compounds are present?
Denote as follows: S = signal, O = organic pollutants, V = volatile solvents, C = chlorinated compounds
2.8.4. Software to detect fraud in consumer phone cards tracks the number of metropolitan areas where calls originate each
day. It is found that 1% of the legitimate users originate calls from two or more metropolitan areas in a single day.
However, 30% of fraudulent users originate calls from two or more metropolitan areas in a single day. The proportion
of fraudulent users is 0.01%. If the same user originates calls from two or more metropolitan areas in a single day, what
is the probability that the user is fraudulent?
2.8.5. Consider the hospital emergency room data given below. Use Bayes’ theorem to calculate the probability
that a person visits hospital 4 given they are LWBS.
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2.8.6. The article “Clinical and Radiographic Outcomes of Four Different Treatment Strategies in Patients with Early
Rheumatoid Arthritis,” [Arthritis & Rheumatism (2005, Vol. 52, pp. 3381–3390)], considered four treatment groups.
The groups consisted of patients with different drug therapies (such as prednisone and infliximab): sequential
monotherapy (group 1), step-up combination therapy (group 2), initial combination therapy
(group 3), or initial combination therapy with infliximab (group 4). Radiographs of hands and feet were used to
evaluate disease progression. The number of patients without progression of joint damage was 76 of 114 patients
(67%), 82 of 112 patients (73%), 104 of 120 patients (87%), and 113 of 121 patients (93%) in groups 1–4, respectively.
Suppose that a patient is selected randomly. Let A denote the event that the patient is in group 1, and let B denote the
event that there is no progression.
Determine the following probabilities:
(a) P (B) (b) P (B | A) (c) P (A | B)
2.8.7. Two Web colors are used for a site advertisement. If a site visitor arrives from an affiliate, the probabilities of the blue
or green colors being used in the advertisement are 0.8 and 0.2, respectively. If the site visitor arrives from a search
site, the probabilities of blue and green colors in the advertisement are 0.4 and 0.6, respectively. The proportions of
visitors from affiliates and search sites are 0.3 and 0.7, respectively. What is the probability that a visitor is from a
search site given that the blue ad was viewed?
2.8.8. A recreational equipment supplier finds that among orders that include tents, 40% also include sleeping mats. Only 5%
of orders that do not include tents do include sleeping mats. Also, 20% of orders include tents. Determine the following
probabilities:
(a) The order includes sleeping mats.
(b) The order includes a tent given it includes sleeping mats.
SM: Sleeping Mats; T: Tents;
P(SM | T) = 0.4; P(SM | T) = 0.05; P(T) = 0.2
2.8.9. An e-mail filter is planned to separate valid e-mails from spam. The word free occurs in 60% of the spam messages and
only 4% of the valid messages. Also, 20% of the messages are spam. Determine the following probabilities:
(a) The message contains free.
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(b) The message is spam given that it contains free.
(c) The message is valid given that it does not contain free.
F: Free; S: Spam; V: Valid
P(F | S) = 0.6, P(F | V) = 0.04
Section 2-9
2.9.1. Decide whether a discrete or continuous random variable is the best model for each of the following variables:
(a) The number of cracks exceeding one-half inch in 10 miles of an interstate highway.
(b) The weight of an injection-molded plastic part.
(c) The number of molecules in a sample of gas.
(d) The concentration of output from a reactor.
(e) The current in an electronic circuit.
2.9.2. Decide whether a discrete or continuous random variable is the best model for each of the following variables:
(a) The time until a projectile returns to earth.
(b) The number of times a transistor in a computer memory changes state in one operation.
(c) The volume of gasoline that is lost to evaporation during the filling of a gas tank.
(d) The outside diameter of a machined shaft.
2.9.3. Decide whether a discrete or continuous random variable is the best model for each of the following variables:
(a) The time for a computer algorithm to assign an image to a category.
(b) The number of bytes used to store a file in a computer.
(c) The ozone concentration in micrograms per cubic meter.
(d) The ejection fraction (volumetric fraction of blood pumped from a heart ventricle with each beat).
(e) The fluid flow rate in liters per minute.
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Supplemental Exercises
2.S4. Samples of laboratory glass are in small, light packaging or heavy, large packaging. Suppose that 2% and 1%,
respectively, of the sample shipped in small and large packages, respectively, break during transit. If 60% of the
samples are shipped in large packages and 40% are shipped in small packages, what proportion of samples break
during shipment?
2.S5. 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
Good
Surface Finish
Excellent
80
2
Good
10
8
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. If a part 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)
Let A = excellent surface finish; B = excellent length
2.S6. A lot contains 15 castings from a local supplier and 25 castings from a supplier in the next state. Two castings are
selected randomly, without replacement, from the lot of 40. Let A be the event that the first casting selected is from the
local supplier, and let B denote the event that the second casting is selected from the local supplier. Determine:
(a) P(A) (b) P(B | A) (c) P(A B) (d) P(A B)
Suppose that 3 castings are selected at random, without replacement, from the lot of 40. In addition to the definitions of
events A and B, let C denote the event that the third casting selected is from the local supplier. Determine:
(e) P(A B C) (f) P(A B C)
2.S7. If A, B, and C are mutually exclusive events, is it possible for P(A) = 0.3, P(B) = 0.4, and P(C) = 0.5? Why or
why not?
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2.S8. Incoming calls to a customer service center are classified as complaints (75% of calls) or requests for information
(25% of calls). Of the complaints, 40% deal with computer equipment that does not respond and 57% deal with
incomplete software installation; in the remaining 3% of complaints, the user has improperly followed the installation
instructions. The requests for information are evenly divided on technical questions (50%) and requests to purchase
more products (50%).
(a) What is the probability that an incoming call to the customer service center will be from a customer who has not
followed installation instructions properly?
(b) Find the probability that an incoming call is a request for purchasing more products.
Let U denote the event that the user has improperly followed installation instructions.
Let C denote the event that the incoming call is a complaint.
Let P denote the event that the incoming call is a request to purchase more products.
Let R denote the event that the incoming call is a request for information.
2.S9. In the manufacturing of a chemical adhesive, 3% of all batches have raw materials from two different lots. This
occurs when holding tanks are replenished and the remaining portion of a lot is insufficient to fill the tanks.
Only 5% of batches with material from a single lot require reprocessing. However, the viscosity of batches consisting
of two or more lots of material is more difficult to control, and 40% of such batches require additional processing to
achieve the required viscosity.
Let A denote the event that a batch is formed from two different lots, and let B denote the event that a lot requires
additional processing. Determine the following probabilities:
(a) P(A) (b) P(A) (c) P(B | A) (d) P(B | A)
(e) P(A B) (f) P(A B) (g) P(B)
2.S10. Semiconductor lasers used in optical storage products require higher power levels for write operations than for read
operations. High-power-level operations lower the useful life of the laser. Lasers in products used for backup of higher-
speed magnetic disks primarily write, and the probability that the useful life exceeds five years is 0.95. Lasers that are
in products that are used for main storage spend approximately an equal amount of time reading and writing, and the
probability that the useful life exceeds five years is 0.995. Now, 25% of the products from a manufacturer are used for
backup and 75% of the products are used for main storage.
Let A denote the event that a laser’s useful life exceeds five years, and let B denote the event that a laser is in a product
that is used for backup.
Use a tree diagram to determine the following:
(a) P(B) (b) P(A | B) (c) P(A | B)
(d) P(A B) (e) P(A B) (f) P(A)
(g) What is the probability that the useful life of a laser exceeds five years?
(h) What is the probability that a laser that failed before five years came from a product used for backup?
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2.S11. A congested computer network has a 0.002 probability of losing a data packet, and packet losses are
independent events. A lost packet must be resent.
(a) What is the probability that an e-mail message with 100 packets will need to be resent?
(b) What is the probability that an e-mail message with 3 packets will need exactly 1 to be resent?
(c) If 10 e-mail messages are sent, each with 100 packets, what is the probability that at least 1 message will need
some packets to be resent?
2.S12. An electronic storage device uses an error recovery procedure that requires an immediate satisfactory readback
of any written data. If the readback is not successful after three writing operations, that sector of the device is
eliminated as unacceptable for data storage. On an acceptable portion of the device, the probability of a satisfactory
readback is 0.98. Assume the readbacks are independent. What is the probability that an acceptable portion of the
device is eliminated as unacceptable for data storage?
2.S13. Energy released from cells breaks the molecular bond and converts ATP (adenosine triphosphate) into ADP (adenosine
diphosphate). Storage of ATP in muscle cells (even for an athlete) can sustain maximal muscle power only for less than
five seconds (a short dash). Three systems are used to replenish ATP—phosphagen system, glycogen-lactic acid system
(anaerobic), and aerobic respiration—but the first is useful only for less than 10 seconds, and even the second system
provides less than two minutes of ATP. An endurance athlete needs to perform below the anaerobic threshold to sustain
energy for extended periods. A sample of 100 individuals is described by the energy system used in exercise at
different intensity levels.
Primarily Aerobic
Period
Yes
No
1
50
7
2
13
30
Applied Statistics and Probability for Engineers, 7th edition 2017
Let A denote the event that an individual is in period 2, and let B denote the event that the energy is primarily aerobic.
Determine the number of individuals in:
(a) A B (b) B (c) A B
2.S14. The probability that a customer’s order is not shipped on time is 0.05. A particular customer places three
orders, and the orders are placed far enough apart in time that they can be considered to be independent events.
(a) What is the probability that all are shipped on time?
(b) What is the probability that exactly one is not shipped on time?
(c) What is the probability that two or more orders are not shipped on time?
Let Ai denote the event that the ith order is shipped on time.
2.S15. In circuit testing of printed circuit boards, each board either fails or does not fail the test. A board that fails the test is
then checked further to determine which one of five defect types is the primary failure mode. Represent the sample
space for this experiment.
2.S16. Transactions to a computer database are either new items or changes to previous items. The addition of an
item can be completed in less than 100 milliseconds 90% of the time, but only 20% of changes to a previous item can
be completed in less than this time. If 30% of transactions are changes, what is the probability that a transaction can be
completed in less than 100 milliseconds?
2-41
2.S17. Let E1, E2, and E3 denote the samples that conform to a percentage of solids specification, a molecular weight
specification, and a color specification, respectively. A total of 240 samples are classified by the E1, E2, and E3
specifications, where yes indicates that the sample conforms.
E3 yes
E2
Yes
No
Total
E1
Yes
200
1
201
No
5
4
9
Total
205
5
210
E3 no
E2
Yes
No
Total
E1
Yes
20
4
24
No
6
0
6
Total
26
4
30
(a) Are E1, E2, and E3 mutually exclusive events?
(b) Are E1, E2, and E3 mutually exclusive events?
(c) What is P(E1 or E2 or E3)?
(d) What is the probability that a sample conforms to all three specifications?
(e) What is the probability that a sample conforms to the E1 or E3 specification?
(f) What is the probability that a sample conforms to the E1, E2, or E3 specification?
2.S18. The following circuit operates if and only if there is a path of functional devices from left to right. Assume devices fail
independently and that the probability of failure of each device is as shown. What is the probability that the circuit
operates?
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2.S19. A steel plate contains 20 bolts. Assume that 5 bolts are not torqued to the proper limit. 4 bolts are selected at random,
without replacement, to be checked for torque.
(a) What is the probability that all 4 of the selected bolts are torqued to the proper limit?
(b) What is the probability that at least 1 of the selected bolts is not torqued to the proper limit?
Let Ai denote the event that the ith bolt selected is not torqued to the proper limit.
2.S20. The British government has stepped up its information campaign regarding foot-and-mouth disease by mailing
brochures to farmers around the country. It is estimated that 99% of Scottish farmers who receive the brochure possess
enough information to deal with an outbreak of the disease, but only 90% of those without the brochure can deal with
an outbreak. After the first three months of mailing, 95% of the farmers in Scotland had received the informative
brochure. Compute the probability that a randomly selected farmer will have enough information to deal effectively
with an outbreak of the disease.
2.S21. It is known that two defective cellular phones were erroneously sent to a shipping lot that now has a total of
75 phones. A sample of phones will be selected from the lot without replacement.
(a) If three phones are inspected, determine the probability that exactly one of the defective phones will be found.
(b) If three phones of the software are inspected, determine the probability that both defective phones will be found.
(c) If 73 phones are inspected, determine the probability that both defective phones will be found. (Hint: Work with the
phones that remain in the lot.)
D = defective copy
2.S22. An encryption-decryption system consists of three elements: encode, transmit, and decode. A faulty encode occurs in
0.5% of the messages processed, transmission errors occur in 1% of the messages, and a decode error occurs in 0.1% of
the messages. Assume the errors are independent.
(a) What is the probability of a completely defect-free message?
(b) What is the probability of a message that has either an encode or a decode error?
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2.S23. The following circuit operates if and only if there is a path of functional devices from left to right. Assume that
devices fail independently and that the probability of failure of each device is as shown. What is the probability that the
circuit does not operate?
2.S24. A robotic insertion tool contains 10 primary components. The probability that any component fails during the
warranty period is 0.01. Assume that the components fail independently and that the tool fails if any component fails.
What is the probability that the tool fails during the warranty period?
2.S25. A machine tool is idle 15% of the time. You request immediate use of the tool on five different occasions during the
year. Assume that your requests represent independent events.
(a) What is the probability that the tool is idle at the time of all of your requests?
(b) What is the probability that the machine is idle at the time of exactly four of your requests?
(c) What is the probability that the tool is idle at the time of at least three of your requests?
2.S26. A company that tracks the use of its Web site determined that the more pages a visitor views, the more likely the visitor
is to request more information (RMI). Use the following table to answer the questions:
Number of pages viewed:
1
2
3
4 or more
Percentage of visitors:
40
30
20
10
Percentage who RMI:
10
10
20
40
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(a) What is the probability that a visitor to the Web site provides contact information?
(b) If a visitor provides contact information, what is the probability that the visitor viewed four or more pages?
2.S27. An article in Genome Research [“An Assessment of Gene Prediction Accuracy in Large DNA Sequences” (2000,
Vol. 10, pp. 1631–1642)] considered the accuracy of commercial software to predict nucleotides in gene sequences.
The following table shows the number of sequences for which the programs produced predictions and the number of
nucleotides correctly predicted (computed globally from the total number of prediction successes and failures on all
sequences).
Number of Sequences
Proportion
GenScan
177
0.93
Blastx default
175
0.91
Blastx topcomboN
174
0.97
Blastx 2 stages
175
0.90
GeneWise
177
0.98
Procrustes
177
0.93
Assume the prediction successes and failures are independent among the programs.
(a) What is the probability that all programs predict a nucleotide correctly?
(b) What is the probability that all programs predict a nucleotide incorrectly?
(c) What is the probability that at least one Blastx program predicts a nucleotide correctly?
2.S28. A batch contains 36 bacterial cells. Assume that 12 of the cells are not capable of cellular replication. Of the cells,
6 are selected at random, without replacement, to be checked for replication.
(a) What is the probability that all 6 of the selected cells are able to replicate?
(b) What is the probability that at least 1 of the selected cells is not capable of replication?
2.S29. A computer system uses passwords that are exactly seven characters, and each character is one of the 26 letters
(a–z) or 10 integers (0–9). Uppercase letters are not used.
(a) How many passwords are possible?
(b) If a password consists of exactly 6 letters and 1 number, how many passwords are possible?
(c) If a password consists of 5 letters followed by 2 numbers, how many passwords are possible?
2.S30. 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
Applied Statistics and Probability for Engineers, 7th edition 2017
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. Suppose a patient is selected randomly.
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 treated with ribavirin plus interferon alfa or interferon alfa, and let B denote
the event that the response is complete. Determine the following probabilities.
2.S31. The article “Clinical and Radiographic Outcomes of Four Different Treatment Strategies in Patients with Early
Rheumatoid Arthritis,” [Arthritis & Rheumatism (2005, Vol. 52, pp. 3381–3390)], considered four treatment groups.
The groups consisted of patients with different drug therapies (such as prednisone and infliximab): sequential
monotherapy (group 1), step-up combination therapy (group 2), initial combination therapy
(group 3), or initial combination therapy with infliximab (group 4). Radiographs of hands and feet were used to
evaluate disease progression. The number of patients without progression of joint damage was 76 of 114 patients
(67%), 82 of 112 patients (73%), 104 of 120 patients (87%), and 113 of 121 patients (93%) in groups 1–4, respectively.
Suppose a patient is selected randomly. Let A denote the event that the patient is in group 1 or 2, and let B denote the
event that there is no progression. Determine the following probabilities:
(a) P(A | B) (b) P(B | A) (c) P(A B) (d) P(A B)