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2-8 RANDOM VARIABLES 53

EXERCISES FOR SECTION 2-7
2-94. Suppose that P1A ƒ B2 0.7, P1A2 0.5, and ucts received good reviews, and 10% of poor products
received good reviews. In addition, 40% of products have
P1B2 0.2. Determine P1B ƒ A2.
been highly successful, 35% have been moderately
2-95. Software to detect fraud in consumer phone cards successful, and 25% have been poor products.
tracks the number of metropolitan areas where calls origi- (a) What is the probability that a product attains a good
nate each day. It is found that 1% of the legitimate users review?
originate calls from two or more metropolitan areas in a (b) If a new design attains a good review, what is the proba-
single day. However, 30% of fraudulent users originate bility that it will be a highly successful product?
calls from two or more metropolitan areas in a single day. (c) If a product does not attain a good review, what is the
The proportion of fraudulent users is 0.01%. If the probability that it will be a highly successful product?
same user originates calls from two or more metropolitan
areas in a single day, what is the probability that the user is 2-98. An inspector working for a manufacturing company
fraudulent? has a 99% chance of correctly identifying defective items and
a 0.5% chance of incorrectly classifying a good item as defec-
2-96. Semiconductor lasers used in optical storage products tive. The company has evidence that its line produces 0.9% of
require higher power levels for write operations than for read nonconforming items.
operations. High-power-level operations lower the useful life (a) What is the probability that an item selected for inspection
of the laser. is classified as defective?
Lasers in products used for backup of higher speed mag- (b) If an item selected at random is classified as nondefective,
netic disks primarily write, and the probability that the useful what is the probability that it is indeed good?
life exceeds five years is 0.95. Lasers that are in products that
are used for main storage spend approximately an equal 2-99. A new analytical method to detect pollutants in water
amount of time reading and writing, and the probability that is being tested. This new method of chemical analysis is im-
the useful life exceeds five years is 0.995. Now, 25% of the portant because, if adopted, it could be used to detect three dif-
products from a manufacturer are used for backup and 75% of ferent contaminants—organic pollutants, volatile solvents,
the products are used for main storage. and chlorinated compounds—instead of having to use a single
Let A denote the event that a laser’s useful life exceeds five test for each pollutant. The makers of the test claim that it can
years, and let B denote the event that a laser is in a product that detect high levels of organic pollutants with 99.7% accuracy,
is used for backup. volatile solvents with 99.95% accuracy, and chlorinated com-
pounds with 89.7% accuracy. If a pollutant is not present, the
Use a tree diagram to determine the following:
test does not signal. Samples are prepared for the calibration
(a) P1B2 (b) P1A ƒ B2
of the test and 60% of them are contaminated with organic
(c) P1A ƒ B¿2 (d) P1A ¨ B2
pollutants, 27% with volatile solvents, and 13% with traces of
(e) P1A ¨ B¿2 (f) P1A2
chlorinated compounds.
(g) What is the probability that the useful life of a laser
A test sample is selected randomly.
exceeds five years?
(a) What is the probability that the test will signal?
(h) What is the probability that a laser that failed before five
(b) If the test signals, what is the probability that chlori-
years came from a product used for backup?
nated compounds are present?
2-97. Customers are used to evaluate preliminary product
designs. In the past, 95% of highly successful products
received good reviews, 60% of moderately successful prod-
2-8 RANDOM VARIABLES

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 suf-
ficient, but in other cases, it is useful to associate a number with each outcome in the sam-
ple 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.