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56 CHAPTER 2 PROBABILITY

device is as shown. What is the probability that the circuit (c) If 73 copies are inspected, determine the probability that
operates? both copies will be found. Hint: Work with the copies that
remain in the lot.
2-114. A robotic insertion tool contains 10 primary compo-
0.01 0.01 0.1
nents. The probability that any component fails during the
warranty period is 0.01. Assume that the components fail
independently and that the tool fails if any component fails.
0.1 0.1 What is the probability that the tool fails during the warranty
period?
2-115. An e-mail message can travel through one of two
2-109. The probability of getting through by telephone to
server routes. The probability of transmission error in each of
buy concert tickets is 0.92. For the same event, the probability
the servers and the proportion of messages that travel each
of accessing the vendor’s Web site is 0.95. Assume that these
route are shown in the following table. Assume that the
two ways to buy tickets are independent. What is the proba-
servers are independent.
bility that someone who tries to buy tickets through the
Internet and by phone will obtain tickets?
probability of error
2-110. The British government has stepped up its information
percentage
campaign regarding foot and mouth disease by mailing
of messages server 1 server 2 server 3 server 4
brochures to farmers around the country. It is estimated that 99%
route 1 30 0.01 0.015
of Scottish farmers who receive the brochure possess enough in-
formation to deal with an outbreak of the disease, but only 90% route 2 70 0.02 0.003
of those without the brochure can deal with an outbreak. After
the first three months of mailing, 95% of the farmers in (a) What is the probability that a message will arrive without
Scotland received the informative brochure. Compute the prob- error?
ability that a randomly selected farmer will have enough infor- (b) If a message arrives in error, what is the probability it was
mation to deal effectively with an outbreak of the disease. sent through route 1?
2-111. In an automated filling operation, the probability of 2-116. A machine tool is idle 15% of the time. You request
an incorrect fill when the process is operated at a low speed is immediate use of the tool on five different occasions during
0.001. When the process is operated at a high speed, the prob- the year. Assume that your requests represent independent
ability of an incorrect fill is 0.01. Assume that 30% of the events.
containers are filled when the process is operated at a high (a) What is the probability that the tool is idle at the time of all
speed and the remainder are filled when the process is of your requests?
operated at a low speed. (b) What is the probability that the machine is idle at the time
(a) What is the probability of an incorrectly filled container? of exactly four of your requests?
(b) If an incorrectly filled container is found, what is the proba- (c) What is the probability that the tool is idle at the time of at
bility that it was filled during the high-speed operation? least three of your requests?
2-112. An encryption-decryption system consists of three 2-117. A lot of 50 spacing washers contains 30 washers that
elements: encode, transmit, and decode. A faulty encode are thicker than the target dimension. Suppose that three wash-
occurs in 0.5% of the messages processed, transmission errors ers are selected at random, without replacement, from the lot.
occur in 1% of the messages, and a decode error occurs in (a) What is the probability that all three washers are thicker
0.1% of the messages. Assume the errors are independent. than the target?
(a) What is the probability of a completely defect-free (b) What is the probability that the third washer selected is
message? thicker than the target if the first two washers selected are
(b) What is the probability of a message that has either an thinner than the target?
encode or a decode error? (c) What is the probability that the third washer selected is
2-113. It is known that two defective copies of a commercial thicker than the target?
software program were erroneously sent to a shipping lot that 2-118. Continuation of Exercise 2-117. Washers are se-
has now a total of 75 copies of the program. A sample of copies lected from the lot at random, without replacement.
will be selected from the lot without replacement. (a) What is the minimum number of washers that need to be
(a) If three copies of the software are inspected, determine the selected so that the probability that all the washers are
probability that exactly one of the defective copies will be thinner than the target is less than 0.10?
found. (b) What is the minimum number of washers that need to be
(b) If three copies of the software are inspected, determine the selected so that the probability that one or more washers
probability that both defective copies will be found. are thicker than the target is at least 0.90?