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58 CHAPTER 2 PROBABILITY

MIND-EXPANDING EXERCISES

2-123. The alignment between the magnetic tape and How many kits should be produced each week to maxi-
head in a magnetic tape storage system affects the per- mize the mean earnings of the firm?
formance of the system. Suppose that 10% of the read 2-126. Assume the following characteristics of the
operations are degraded by skewed alignments, 5% by inspection process in Exercise 2-107. If an operator
off-center alignments, 1% by both skewness and off- checks a bolt, the probability that an incorrectly
center, and the remaining read operations are properly torqued bolt is identified is 0.95. If a checked bolt is
aligned. The probability of a read error is 0.01 from a correctly torqued, the operator’s conclusion is always
skewed alignment, 0.02 from an off-center alignment, correct. What is the probability that at least one bolt in
0.06 from both conditions, and 0.001 from a proper the sample of four is identified as being incorrectly
alignment. What is the probability of a read error. torqued?
2-124. Suppose that a lot of washers is large enough 2-127. If the events A and B are independent, show
that it can be assumed that the sampling is done with re- that A¿ and B¿ are independent.
placement. Assume that 60% of the washers exceed the 2-128. Suppose that a table of part counts is generalized
target thickness. as follows:
(a) What is the minimum number of washers that need
conforms
to be selected so that the probability that all the
washers are thinner than the target is less than 0.10? yes no
(b) What is the minimum number of washers that need to supplier 1 ka kb
be selected so that the probability that one or more 2 a b
washers are thicker than the target is at least 0.90? where a, b, and k are positive integers. Let A denote the
2-125. A biotechnology manufacturing firm can pro- event that a part is from supplier 1 and let B denote the
duce diagnostic test kits at a cost of $20. Each kit for event that a part conforms to specifications. Show that
which there is a demand in the week of production can be A and B are independent events.
sold for $100. However, the half-life of components in This exercise illustrates the result that whenever the
the kit requires the kit to be scrapped if it is not sold in rows of a table (with r rows and c columns) are propor-
the week of production. The cost of scrapping the kit is tional, an event defined by a row category and an event
$5. The weekly demand is summarized as follows: defined by a column category are independent.
weekly demand
Number of
units 0 50 100 200
Probability of
demand 0.05 0.4 0.3 0.25

IMPORTANT TERMS AND CONCEPTS
In the E-book, click on any Event Random variables CD MATERIAL
term or concept below to Independence discrete and
Permutation
go to that subject. Multiplication rule continuous
Combination
Addition rule Mutually exclusive Sample spaces—discrete
Axioms of probability events and continuous
Bayes’ theorem Outcome Total probability rule
Conditional probability Random experiment With or without
Equally likely outcomes replacement