Unformatted text preview: 58 CHAPTER 2 PROBABILITY facultative anaerobes, 21% contain microaerophiles, 13% contain nanaerobes, and 8% contain aerotolerant. A test sample is selected randomly. (a) What is the probability that the process will signal? (b) If the test signals, what is the probability that mi-
croaerophiles are present? 2—120. In the 2004 presidential election, exit polls from
the critical state of Ohio provided the following results: Bush Kerry
no college degree (62%) 50% 50%
college graduate (38%) 53% 46% If a randomly selected respondent voted for Bush, what is
the probability that the person has a college degree? 2—121. Customers are used to evaluate preliminary product designs. In the past, 95% of highly successful products received good reviews, 60% of moderately successful products received good reviews, and 10% of poor products received good reviews. In addition, 40% of products have been highly successful, 35% have been moderately successful, and 25% have been poor products. (a) What is the probability that a product attains a good
review? (b) If a new design attains a good review, what is the proba-
bility that it will be a highly successﬁil product? 2—8 RANDOM VARIABLES (c) If a product does not attain a good review, what is the
probability that it will be a higth successful product? 2,122. An inspector working for a manufacturing company
has a 99% chance of correctly identifying defective items and
a 0.5% chance of incorrectly classifying a good item as defec-
tive. The company has evidence that its line produces 0.9% of
nonconforming items. '
(a) What is the probability that an item selected for inspection
is classiﬁed as defective?
(b) If an item selected at random is classiﬁed as nondefective,
what is the probability that it is indeed good? 2—123. A new analytical method to detect pollutants in wa-
ter is being tested. This new method of chemical analysis is
important because, if adopted, it could be used to detect three
different contaminants—organic pollutants, volatile solvents,
and chlorinated compounds—instead of having to use a single
test for each pollutant. The makers of the test claim that it can
detect high levels of organic pollutants with 99.7% accuracy,
volatile solvents with 99.95% accuracy, and chlorinated com-
pounds with 89.7% accuracy. If a pollutant is not present, the
test does not signal. Samples are prepared for the calibration
of the test and 60% of them are contaminated with organic _
pollutants, 27% with volatile solvents, and 13% with traces of A
chlorinated compounds. A test sample is selected randomly.
(a) What is the probability that the test will signal? (b) If the test signals, what is the probability that chlori- nated compounds are present? We often summarize the outcome from a random experiment by a simple number. In many of
the examples of random experiments that we have considered, the sample space has been a
description of possible outcomes. In some cases, descriptions of outcomes are sufﬁcient, but
in other cases, it is useful to associate a number with each outcome in the sample space.
Because the particular outcome of the experiment is not known in advance, the resulting
value of our variable is not known in advance. For this reason, the variable that associates a
number with the outcome of a random experiment is referred to as a random variable. Random
Variable A random variable is a function that assigns a real number to each outcome in the . z sample space of a random experiment. _ , , 5‘ Notation is used to distinguish between a random variable and the real number. A random variable is denoted by an uppercase letter such as X. After an experiment __ is conducted, the measured value of the random variable is denoted byra IoWercase; :1
letter such as x = 70 milliamperes. ' ...
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- Spring '08