Page 50 - Applied statistics and probability for engineers

28 Chapter 2/Probability

2-29. The rise time of a reactor is measured in minutes 2-38. A manufacturing process consists of 10 operations
(and fractions of minutes). Let the sample space be posi- that can be completed in any order. How many different pro-
tive, real numbers. Deine the events A and B as follows: duction sequences are possible?
. }
. }
A = { x x < 72 5 and B = { x x > 52 5 . 2-39. A manufacturing operation consists of 10 opera-
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Describe each of the following events: tions. However, ive machining operations must be completed
(a) A′ (b) B′ before any of the remaining ive assembly operations can begin.
(c) A∩ B (d) A∪ B Within each set of ive, operations can be completed in any
2-30. A sample of two items is selected without replacement order. How many different production sequences are possible?
from a batch. Describe the (ordered) sample space for each of the 2-40. In a sheet metal operation, three notches and four
following batches: bends are required. If the operations can be done in any order,
(a) The batch contains the items { , , , }a b c d . how many different ways of completing the manufacturing are
(b) The batch contains the items { , , , , , , }a b c d e f g . possible?
(c) The batch contains 4 defective items and 20 good items. 2-41. A batch of 140 semiconductor chips is inspected by
(d) The batch contains 1 defective item and 20 good items. choosing a sample of 5 chips. Assume 10 of the chips do not
2-31. A sample of two printed circuit boards is selected conform to customer requirements.
without replacement from a batch. Describe the (ordered) (a) How many different samples are possible?
sample space for each of the following batches: (b) How many samples of ive contain exactly one noncon-
(a) The batch contains 90 boards that are not defective, 8 forming chip?
boards with minor defects, and 2 boards with major defects. (c) How many samples of ive contain at least one noncon-
(b) The batch contains 90 boards that are not defective, 8 forming chip?
boards with minor defects, and 1 board with major defects. 2-42. In the layout of a printed circuit board for an electronic
2-32. Counts of the Web pages provided by each of two com- product, 12 different locations can accommodate chips.
puter servers in a selected hour of the day are recorded. Let A (a) If ive different types of chips are to be placed on the board,
denote the event that at least 10 pages are provided by server 1, how many different layouts are possible?
and let B denote the event that at least 20 pages are provided by (b) If the ive chips that are placed on the board are of the same
server 2. Describe the sample space for the numbers of pages type, how many different layouts are possible?
for the two servers graphically in an x − y plot. Show each of 2-43. In the laboratory analysis of samples from a chemical
the following events on the sample space graph: process, ive samples from the process are analyzed daily. In
(a) A (b) B addition, a control sample is analyzed twice each day to check
(c) A∩ B (d) A∪ B the calibration of the laboratory instruments.
(a) How many different sequences of process and control sam-
2-33. A reactor’s rise time is measured in minutes (and frac-
ples are possible each day? Assume that the ive process
tions of minutes). Let the sample space for the rise time of each
samples are considered identical and that the two control
batch be positive, real numbers. Consider the rise times of two
samples are considered identical.
batches. Let A denote the event that the rise time of batch 1 is
(b) How many different sequences of process and control sam-
less than 72.5 minutes, and let B denote the event that the rise
ples are possible if we consider the ive process samples to
time of batch 2 is greater than 52.5 minutes.
be different and the two control samples to be identical?
Describe the sample space for the rise time of two batches
(c) For the same situation as part (b), how many sequences
graphically and show each of the following events on a two-
are possible if the irst test of each day must be a control
dimensional plot:
(a) A (b) B′ sample?
(c) A∩ B (d) A∪ B 2-44. In the design of an electromechanical product, 12 com-
2-34. A wireless garage door opener has a code deter- ponents are to be stacked into a cylindrical casing in a manner
mined by the up or down setting of 12 switches. How many that minimizes the impact of shocks. One end of the casing is
outcomes are in the sample space of possible codes? designated as the bottom and the other end is the top.
2-35. An order for a computer can specify any one of ive (a) If all components are different, how many different designs
memory sizes, any one of three types of displays, and any one are possible?
of four sizes of a hard disk, and can either include or not include (b) If seven components are identical to one another, but
a pen tablet. How many different systems can be ordered? the others are different, how many different designs are
2-36. In a manufacturing operation, a part is produced by possible?
machining, polishing, and painting. If there are three machine (c) If three components are of one type and identical to one
tools, four polishing tools, and three painting tools, how many another, and four components are of another type and
different routings (consisting of machining, followed by pol- identical to one another, but the others are different, how
ishing, and followed by painting) for a part are possible? many different designs are possible?
2-37. New designs for a wastewater treatment tank have 2-45. Consider the design of a communication system.
proposed three possible shapes, four possible sizes, three loca- (a) How many three-digit phone preixes that are used to repre-
tions for input valves, and four locations for output valves. How sent a particular geographic area (such as an area code) can
many different product designs are possible? be created from the digits 0 through 9?