Page 51 - Applied statistics and probability for engineers

Section 2-1/Sample Spaces and Events 29

(b) As in part (a), how many three-digit phone preixes are possible (a) A∩ B (b) A′ (c) A∪ B (d) A∪ B′ (e) A′ ∩ B′
that do not start with 0 or 1, but contain 0 or 1 as the middle digit? 2-53. An article in The Journal of Data Science [“A Statistical
(c) How many three-digit phone preixes are possible in which Analysis of Well Failures in Baltimore County” (2009, Vol. 7,
no digit appears more than once in each preix? pp. 111–127)] provided the following table of well failures for
2-46. A byte is a sequence of eight bits and each bit is either 0 or 1. different geological formation groups in Baltimore County.
(a) How many different bytes are possible?
(b) If the irst bit of a byte is a parity check, that is, the irst byte Wells
is determined from the other seven bits, how many different Geological Formation Group Failed Total
bytes are possible? Gneiss 170 1685
2-47. In a chemical plant, 24 holding tanks are used for inal Granite 2 28
product storage. Four tanks are selected at random and without Loch raven schist 443 3733
replacement. Suppose that six of the tanks contain material in Maic 14 363
which the viscosity exceeds the customer requirements. Marble 29 309
(a) What is the probability that exactly one tank in the sample
contains high-viscosity material? Prettyboy schist 60 1403
(b) What is the probability that at least one tank in the sample Other schists 46 ‹3
contains high-viscosity material? Serpentine 3 39
(c) In addition to the six tanks with high viscosity levels, four Let A denote the event that the geological formation has more
different tanks contain material with high impurities. What than 1000 wells, and let B denote the event that a well failed.
is the probability that exactly one tank in the sample con- Determine the number of wells in each of the following events.
tains high-viscosity material and exactly one tank in the (a) A∩ B (b) A′ (c) A∪ B (d) A∩ B′ (e) A′ ∩ B′
sample contains material with high impurities?
2-54. Similar to the hospital schedule in Example 2-11, sup-
2-48. Plastic parts produced by an injection-molding pose that an operat ing room needs to handle three knee, four
operation are checked for conformance to speciications. Each hip, and ive shoulder surgeries.
tool contains 12 cavities in which parts are produced, and these (a) How many different sequences are possible?
parts fall into a conveyor when the press opens. An inspector (b) How many different sequences have all hip, knee, and
chooses 3 parts from among the 12 at random. Two cavities are shoulder surgeries scheduled consecutively?
affected by a temperature malfunction that results in parts that (c) How many different schedules begin and end with a knee
do not conform to speciications.
surgery?
(a) How many samples contain exactly 1 nonconforming part?
(b) How many samples contain at least 1 nonconforming part? 2-55. Consider the bar code in Example 2-12. One code is still
held back as a delimiter. For each of the following cases, how
2-49. A bin of 50 parts contains 5 that are defective. A sample many characters can be encoded?
of 10 parts is selected at random, without replacement. How (a) The constraint of exactly two wide bars is replaced with
many samples contain at least four defective parts?
one that requires exactly one wide bar.
2-50. The following table summarizes 204 endothermic reac- (b) The constraint of exactly two wide bars is replaced with
tions involving sodium bicarbonate. one that allows either one or two wide bars.
Final Temperature (c) The constraint of exactly two wide bars is dropped.
Conditions Heat Absorbed (cal) (d) The constraints of exactly two wide bars and one wide
space are dropped.
Below Target Above Target
2-56. A computer system uses passwords that contain exactly
266 K 12 40
eight characters, and each character is 1 of the 26 lowercase
271 K 44 16 letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9).
274 K 56 36 Let Ω denote the set of all possible passwords, and let A and
B denote the events that consist of passwords with only letters
Let A denote the event that a reaction’s inal temperature is 271 K or only integers, respectively. Determine the number of pass-
or less. Let B denote the event that the heat absorbed is below words in each of the following events.
target. Determine the number of reactions in each of the (a) Ω (b) A (c) A′ ∩ B′
following events.
(d) Passwords that contain at least 1 integer
(a) A∩ B (b) A′ (c) A∪ B (d) A∪ B′ (e) A′ ∩ B′ (e) Passwords that contain exactly 1 integer
2-51. A Web ad can be designed from four different colors, 2-57. The article “Term Eficacy of Ribavirin Plus Interferon
three font types, ive font sizes, three images, and ive text Alfa in the Treat ment of Chronic Hepatitis C,” [Gastroenterol-
phrases. How many different designs are possible? ogy (1996, Vol. 111, no. 5, pp. 1307–1312)], considered the
2-52. Consider the hospital emergency department data in Exam- effect of two treatments and a control for treatment of hepatitis
ple 2-8. Let A denote the event that a visit is to hospital 1, and let C. The following table provides the total patients in each group
B denote the event that a visit results in admittance to any hospital. and the number that showed a complete (positive) response
Determine the number of persons in each of the following events. after 24 weeks of treatment.