Page 86 - Applied statistics and probability for engineers

64 Chapter 2/Probability

Mind-Expanding Exercises

2-230. Suppose documents in a lending organization are How many kits should be produced each week to maximize
selected randomly (without replacement) for review. In a set the irm’s mean earnings?
of 50 documents, suppose that 2 actually contain errors. 2-233. Assume the following characteristics of the inspec-
(a) What is the minimum sample size such that the probability tion process in Exercise 2-207. If an operator checks a bolt,
exceeds 0.90 that at least 1 document in error is selected? the probability that an incorrectly torqued bolt is identiied
(b) Comment on the effectiveness of sampling inspection to is 0.95. If a checked bolt is correctly torqued, the operator’s
detect errors. conclusion is always correct. What is the probability that
at least one bolt in the sample of four is identiied as being
2-231. Suppose that a lot of washers is large enough that it
incorrectly torqued?
can be assumed that the sampling is done with replacement.
2-234. If the events A and B are independent, show that A′
Assume that 60% of the washers exceed the target thickness.
and B′ are independent.
(a) What is the minimum number of washers that need to
2-235. Suppose that a table of part counts is generalized
be selected so that the probability that none is thicker
as follows:
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 1 or more wash- Conforms
ers are thicker than the target is at least 0.90? Yes No
2-232. A biotechnology manufacturing irm can produce Supplier 1 ka kb
diagnostic test kits at a cost of $20. Each kit for which there 2 a b
is a demand in the week of production can be sold for $100.
However, the half-life of components in the kit requires the where a, b, and k are positive integers. Let A denote the
kit to be scrapped if it is not sold in the week of production. event that a part is from supplier 1, and let B denote the
The cost of scrapping the kit is $5. The weekly demand is event that a part conforms to speciications. Show that A
summarized as follows: and B are independent events.
This exercise illustrates the result that whenever the rows
Weekly Demand of a table (with r rows and c columns) are proportional, an
Number of units 0 50 100 200 event deined by a row category and an event deined by a
column category are independent.
Probability of
demand 0.05 0.4 0.3 0.25

Important Terms and Concepts

Addition rule Equally likely Permutation Simpson’s paradox
Axioms of probability outcomes Probability Total probability rule
Bayes’ theorem Event Random samples Tree diagram
Combination Independence Random variables—discrete Venn diagram
Conditional probability Multiplication rule and continuous With or without replacement
Counting Mutually exclusive events Sample spaces—discrete and
techniques Outcome continuous