Applied Statistics and Probability for Engineers

58 CHAPTER 2 PROBABILITY

2-123. The alignment between the magnetic tape and

head in a magnetic tape storage system affects the per-

formance of the system. Suppose that 10% of the read

operations are degraded by skewed alignments, 5% by

off-center alignments, 1% by both skewness and off-

center, and the remaining read operations are properly

aligned. The probability of a read error is 0.01 from a

skewed alignment, 0.02 from an off-center alignment,

0.06 from both conditions, and 0.001 from a proper

alignment. What is the probability of a read error.

2-124. Suppose that a lot of washers is large enough

that it can be assumed that the sampling is done with re-

placement. Assume that 60% of the washers exceed the

target thickness.

(a) What is the minimum number of washers that need

to be selected so that the probability that all the

washers are thinner than the target is less than 0.10?

(b) What is the minimum number of washers that need to

be selected so that the probability that one or more

washers are thicker than the target is at least 0.90?

2-125. A biotechnology manufacturing firm can pro-

duce diagnostic test kits at a cost of $20. Each kit for

which there is a demand in the week of production can be

sold for $100. However, the half-life of components in

the kit requires the kit to be scrapped if it is not sold in

the week of production. The cost of scrapping the kit is

$5. The weekly demand is summarized as follows:

weekly demand

Number of

units 0 50 100 200

Probability of

demand 0.05 0.4 0.3 0.25

How many kits should be produced each week to maxi-

mize the mean earnings of the firm?

2-126. Assume the following characteristics of the

inspection process in Exercise 2-107. If an operator

checks a bolt, the probability that an incorrectly

torqued bolt is identified is 0.95. If a checked bolt is

correctly torqued, the operator’s conclusion is always

correct. What is the probability that at least one bolt in

the sample of four is identified as being incorrectly

torqued?

2-127. If the events Aand Bare independent, show

that and are independent.

2-128. Suppose that a table of part counts is generalized

as follows:

conforms

yes no

supplier 1 ka kb

2 ab

where a,b, and kare positive integers. Let Adenote the

event that a part is from supplier 1 and let Bdenote the

event that a part conforms to specifications. Show that

Aand Bare independent events.

This exercise illustrates the result that whenever the

rows of a table (with rrows and ccolumns) are propor-

tional, an event defined by a row category and an event

defined by a column category are independent.

A¿ B¿

MIND-EXPANDING EXERCISES

In the E-book, click on any

term or concept below to

go to that subject.

Addition rule

Axioms of probability

Bayes’ theorem

Conditional probability

Equally likely outcomes

Event

Independence

Multiplication rule

Mutually exclusive

events

Outcome

Random experiment

Random variables

discrete and

continuous

Sample spaces—discrete

and continuous

Total probability rule

With or without

replacement

CD MATERIAL

Permutation

Combination

IMPORTANT TERMS AND CONCEPTS

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