Page 67 - Applied Statistics And Probability For Engineers

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50 CHAPTER 2 PROBABILITY

The solution can be obtained from a partition of the graph into three columns.
The probability that there is a path of functional devices only through the three units on the
left can be determined from the independence in a manner similar to the previous example.
It is

1 0.1 3

Similarly, the probability that there is a path of functional devices only through the two units
in the middle is

1 0.05 2

The probability that there is a path of functional devices only through the one unit on the right
is simply the probability that the device functions, namely, 0.99. Therefore, with the inde-
pendence assumption used again, the solution is
3
2
11 0.1 211 0.05 210.992 0.987

EXERCISES FOR SECTION 2-6

2-81. If P1A ƒ B2 0.4, P1B2 0.8, and P1A2 0.5, are conforms
the events A and B independent? yes no
2-82. If P1A ƒ B2 0.3, P1B2 0.8, and P1A2 0.3, are 1 22 8
the events B and the complement of A independent? supplier 2 25 5
2-83. Disks of polycarbonate plastic from a supplier are an- 3 30 10
alyzed for scratch and shock resistance. The results from 100
disks are summarized as follows: Let A denote the event that a sample is from supplier 1, and let
B denote the event that a sample conforms to specifications.
shock resistance
(a) Are events A and B independent?
high low (b) Determine P1B ƒ A2.
scratch high 70 9 2-86. If P1A2 0.2, P1B2 0.2, and A and B are mutually
resistance low 16 5 exclusive, are they independent?
2-87. The probability that a lab specimen contains high lev-
Let A denote the event that a disk has high shock resistance,
and let B denote the event that a disk has high scratch resist- els of contamination is 0.10. Five samples are checked, and
ance. Are events A and B independent? the samples are independent.
(a) What is the probability that none contains high levels of
2-84. Samples of a cast aluminum part are classified on the contamination?
basis of surface finish (in microinches) and length measure- (b) What is the probability that exactly one contains high
ments. The results of 100 parts are summarized as follows:
levels of contamination?
length (c) What is the probability that at least one contains high
levels of contamination?
excellent good
2-88. In a test of a printed circuit board using a random test
surface excellent 80 2
pattern, an array of 10 bits is equally likely to be 0 or 1.
finish good 10 8
Assume the bits are independent.
Let A denote the event that a sample has excellent surface fin- (a) What is the probability that all bits are 1s?
ish, and let B denote the event that a sample has excellent (b) What is the probability that all bits are 0s?
length. Are events A and B independent? (c) What is the probability that exactly five bits are 1s and five
2-85. Samples of emissions from three suppliers are classi- bits are 0s?
fied for conformance to air-quality specifications. The results 2-89. Eight cavities in an injection-molding tool produce
from 100 samples are summarized as follows: plastic connectors that fall into a common stream. A sample is