Page 81 - Applied statistics and probability for engineers

Section 2-8/Random Variables 59

Supplemental Exercises

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
2-185. Samples of laboratory glass are in small, light (d) If a shaft is selected at random, what is the probability that
packaging or heavy, large packaging. Suppose that 2% and 1%, the shaft conforms to surface inish requirements or the
respectively, of the sample shipped in small and large packages, shaft is from tool 2?
respectively, break during transit. If 60% of the samples are shipped 2-189. If A B, , and C are mutually exclusive events, is it possible
4
(
(
3
5
in large packages and 40% are shipped in small packages, what for P A( ) = 0 . , P B) = 0 . , and P C) = 0 . ? Why or why not?
proportion of samples break during shipment? 2-190. The analysis of shafts for a compressor is
2-186. A sample of three calculators is selected from a manufac- summarized by conformance to speciications:
turing line, and each calculator is classiied as either defective or Roundness Conforms
acceptable. Let A B, , and C denote the events that the irst, second,
Yes No
and third calculators, respectively, are defective.
Surface inish Yes 345 5
(a) Describe the sample space for this experiment with a tree
Conforms No 12 8
diagram.
(a) If we know that a shaft conforms to roundness requirements,
Use the tree diagram to describe each of the following events:
(b) A (c) B (d) A∩ B (e) B ∪ C what is the probability that it conforms to surface inish
2-187. Samples of a cast aluminum part are classiied on requirements?
(b) If we know that a shaft does not conform to roundness
the basis of surface inish (in microinches) and edge inish. The
requirements, what is the probability that it conforms to
results of 100 parts are summarized as follows:
surface inish requirements?
Edge Finish
2-191. A researcher receives 100 containers of oxygen.
Excellent Good
Of those containers, 20 have oxygen that is not ionized, and the
Surface Excellent 80 2 rest are ionized. Two samples are randomly selected, without
Finish Good 10 8 replacement, from the lot.
Let A denote the event that a sample has excellent surface in- (a) What is the probability that the irst one selected is not
ish, and let B denote the event that a sample has excellent edge ionized?
inish. If a part is selected at random, determine the following (b) What is the probability that the second one selected is not
probabilities: ionized given that the irst one was ionized?
(a) P A( ) (b) P B( ) (c) P A( )′ (c) What is the probability that both are ionized?
(d) P A( ∩ B) (e) P A( ∪ B) (f) P A( ′ B) (d) How does the answer in part (b) change if samples selected
were replaced prior to the next selection?
2-188. Shafts are classiied in terms of the machine tool that
2-192. A lot contains 15 castings from a local supplier and
was used for manufacturing the shaft and conformance to sur-
25 castings from a supplier in the next state. Two castings are
face inish and roundness.
selected randomly, without replacement, from the lot of 40. Let
Tool 1 Roundness Conforms A be the event that the irst casting selected is from the local
Yes No supplier, and let B denote the event that the second casting is
Surface Finish Yes 200 1 selected from the local supplier. Determine:
| (
(
(
Conforms No 4 2 (a) P A ( ) (b) P B A) (c) P A∩ B) (d) P A∪ B)
Suppose that 3 castings are selected at random, without replace-
Tool 2 Roundness Conforms
ment, from the lot of 40. In addition to the deinitions of events
Yes No A and B, let C denote the event that the third casting selected is
Surface Finish Yes 145 4 from the local supplier. Determine:
(
(
C′
B
B
Conforms No 8 6 (e) P A∩ ∩ C) (f) P A∩ ∩ )
(a) If a shaft is selected at random, what is the probability 2-193. In the manufacturing of a chemical adhesive, 3%
that the shaft conforms to surface inish requirements or to of all batches have raw materials from two different lots. This
roundness requirements or is from tool 1? occurs when holding tanks are replenished and the remaining
(b) If a shaft is selected at random, what is the probability that portion of a lot is insuficient to ill the tanks.
the shaft conforms to surface inish requirements or does not Only 5% of batches with material from a single lot require
conform to roundness requirements or is from tool 2? reprocessing. However, the viscosity of batches consisting of
(c) If a shaft is selected at random, what is the probability that two or more lots of material is more dificult to control, and
the shaft conforms to both surface inish and roundness 40% of such batches require additional processing to achieve
requirements or the shaft is from tool 2? the required viscosity.