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46 CHAPTER 2 PROBABILITY

2-75. The edge roughness of slit paper products increases as (b) Find the probability that an incoming call is a request for
knife blades wear. Only 1% of products slit with new blades purchasing more products.
have rough edges, 3% of products slit with blades of average 2-78. Computer keyboard failures are due to faulty electri-
sharpness exhibit roughness, and 5% of products slit with cal connects (12%) or mechanical defects (88%). Mechanical
worn blades exhibit roughness. If 25% of the blades in manu- defects are related to loose keys (27%) or improper assembly
facturing are new, 60% are of average sharpness, and 15% are (73%). Electrical connect defects are caused by defective
worn, what is the proportion of products that exhibit edge wires (35%), improper connections (13%), or poorly welded
roughness? wires (52%).
2-76. Samples of laboratory glass are in small, light pack- (a) Find the probability that a failure is due to loose keys.
aging or heavy, large packaging. Suppose that 2 and 1% of (b) Find the probability that a failure is due to improperly
the sample shipped in small and large packages, respec- connected or poorly welded wires.
tively, break during transit. If 60% of the samples are 2-79. A batch of 25 injection-molded parts contains 5 that
shipped in large packages and 40% are shipped in small have suffered excessive shrinkage.
packages, what proportion of samples break during (a) If two parts are selected at random, and without replace-
shipment? ment, what is the probability that the second part selected
2-77. Incoming calls to a customer service center are classi- is one with excessive shrinkage?
fied as complaints (75% of call) or requests for information (b) If three parts are selected at random, and without replace-
(25% of calls). Of the complaints, 40% deal with computer ment, what is the probability that the third part selected is
equipment that does not respond and 57% deal with one with excessive shrinkage?
incomplete software installation; and in the remaining 3% of 2-80. A lot of 100 semiconductor chips contains 20 that are
complaints the user has improperly followed the installation defective.
instructions. The requests for information are evenly divided (a) Two are selected, at random, without replacement, from
on technical questions (50%) and requests to purchase more the lot. Determine the probability that the second chip se-
products (50%). lected is defective.
(a) What is the probability that an incoming call to the cus- (b) Three are selected, at random, without replacement,
tomer service center will be from a customer who has not from the lot. Determine the probability that all are
followed installation instructions properly? defective.

2-6 INDEPENDENCE

In some cases, the conditional probability of P1B ƒ A2 might equal P(B). In this special case,
knowledge that the outcome of the experiment is in event A does not affect the probability that
the outcome is in event B.

EXAMPLE 2-23 Suppose a day’s production of 850 manufactured parts contains 50 parts that do not meet
customer requirements. Suppose two parts are selected from the batch, but the first part is
replaced before the second part is selected. What is the probability that the second part is
defective (denoted as B) given that the first part is defective (denoted as A)? The probability
needed can be expressed as P1B ƒ A2.
Because the first part is replaced prior to selecting the second part, the batch still contains
850 parts, of which 50 are defective. Therefore, the probability of B does not depend on
whether or not the first part was defective. That is,

P1B ƒ A2 50
850

Also, the probability that both parts are defective is

50 50
P1A ¨ B2 P1B 0 A2P1A2 a b a b 0.0035
850 850