Page 61 - Applied statistics and probability for engineers

Section 2-3/Addition Rules 39

(a) If a strand is randomly selected, what is the probability that (a) P A ( ) (b) P B ( )
(
(
its conductivity is high and its strength is high? (c) P A∩ B) (d) P A∪ B)
(b) If a strand is randomly selected, what is the probability that
its conductivity is low or its strength is low? 2-91. Consider the endothermic reactions in Exercise 2-50. Let
(c) Consider the event that a strand has low conductivity and the A denote the event that a reaction's inal temperature is 271 K or
event that the strand has low strength. Are these two events less. Let B denote the event that the heat absorbed is above target.
mutually exclusive? Use the addition rules to calculate the following probabilities.
(
(
(
(a) P A∪ B) (b) P A∩ ) (c) P A′ ∪ B′)
B′
2-87. The analysis of shafts for a compressor is summa-
rized by conformance to speciications. 2-92. A Web ad can be designed from four different colors, three
font types, ive font sizes, three images, and ive text phrases. A
Roundness Conforms
speciic design is randomly generated by the Web server when you
Yes No visit the site. Let A denote the event that the design color is red,
Surface Finish Yes 345 5 and let B denote the event that the font size is not the smallest one.
Conforms No 12 8 Use the addition rules to calculate the following probabilities.
(
(
(
B′
(a) If a shaft is selected at random, what is the probability that (a) P A∪ B) (b) P A∪ ) (c) P A′ ∪ B′)
it conforms to surface inish requirements?
2-93. Consider the hospital emergency room data in Example
(b) What is the probability that the selected shaft conforms to
2-8. Let A denote the event that a visit is to hospital 4, and let B
surface inish requirements or to roundness requirements?
denote the event that a visit results in LWBS (at any hospital).
(c) What is the probability that the selected shaft either con-
Use the addition rules to calculate the following probabilities.
(
(
(
forms to surface inish requirements or does not conform (a) P A∪ B) (b) P A∪ ) (c) P A′ ∪ B′)
B′
to roundness requirements?
2-94. Consider the well failure data in Exercise 2-53. Let A
(d) What is the probability that the selected shaft conforms to
denote the event that the geological formation has more than
both surface inish and roundness requirements?
1000 wells, and let B denote the event that a well failed. Use
2-88. Cooking oil is produced in two main varieties: mono-
the addition rules to calculate the following probabilities.
(
(
(
and polyunsaturated. Two common sources of cooking oil are (a) P A∪ B) (b) P A∪ ) (c) P A′ ∪ B′)
B′
corn and canola. The following table shows the number of bot-
tles of these oils at a supermarket: 2-95. Consider the bar code in Example 2-12. Suppose that all
40 codes are equally likely (none is held back as a delimiter).
Type of oil
Determine the probability for each of the following:
Canola Corn (a) The irst bar is wide or the second bar is wide.
Type of Unsaturation Mono 7 13 (b) Neither the irst nor the second bar is wide.
Poly 93 77 (c) The irst bar is wide or the second bar is not wide.
(a) If a bottle of oil is selected at random, what is the probabil- (d) The irst bar is wide or the irst space is wide.
ity that it belongs to the polyunsaturated category? 2-96. Consider the three patient groups in Exercise 2-57. Let
(b) What is the probability that the chosen bottle is monoun- A denote the event that the patient was treated with ribavirin
saturated canola oil? plus interferon alfa, and let B denote the event that the response
2-89. A manufacturer of front lights for automobiles tests was complete. Determine the following probabilities:
lamps under a high-humidity, high-temperature environment (a) P A( ∪ B) (b) P A′ B) (c) P A( ∪ B )
′
(
using intensity and useful life as the responses of interest. The
following table shows the performance of 130 lamps: 2-97. A computer system uses passwords that contain exactly
eight characters, and each character is one of the 26 lowercase
Useful life 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
Satisfactory Unsatisfactory
the events that consist of passwords with only letters or only
Intensity Satisfactory 117 3
integers, respectively. Determine the following probabilities:
Unsatisfactory 8 2 (a) P A( ∪ B) (b) P A( ′ B)
(a) Find the probability that a randomly selected lamp will (c) P (Password contains exactly 1 or 2 integers)
yield unsatisfactory results under any criteria. 2-98. The article [“Clinical and Radiographic Outcomes of Four
(b) The customers for these lamps demand 95% satisfactory Different Treatment Strategies in Patients with Early Rheumatoid
results. Can the lamp manufacturer meet this demand?
Arthritis,” Arthritis & Rheumatism (2005, Vol. 52, pp. 3381–
2-90. A computer system uses passwords that are six 3390)] considered four treatment groups. The groups consisted of
characters, and each character is one of the 26 letters (a–z) or patients with different drug therapies (such as prednisone and inf-
10 integers (0–9). Uppercase letters are not used. Let A denote liximab): sequential monotherapy (group 1), step-up combination
the event that a password begins with a vowel (either a, e, i, o, therapy (group 2), initial combination therapy (group 3), or ini-
or u), and let B denote the event that a password ends with an tial combination therapy with inliximab (group 4). Radiographs
even number (either 0, 2, 4, 6, or 8). Suppose a hacker selects of hands and feet were used to evaluate disease pro gression. The
a password at random. Determine the following probabilities: number of patients without progression of joint damage was 76 of