Page 75 - Applied statistics and probability for engineers

Section 2-6/Independence 53

2-147. Samples of emissions from three suppliers are classi- (a) What is the probability that a player defeats all four oppo-
ied for conformance to air-quality speciications. The results nents in a game?
from 100 samples are summarized as follows: (b) What is the probability that a player defeats at least two
opponents in a game?
Conforms (c) If the game is played three times, what is the probability
Yes No that the player defeats all four opponents at least once?
1 22 8 2-153. In an acid-base titration, a base or acid is gradually
added to the other until they have completely neutralized each
Supplier 2 25 5
other. Because acids and bases are usually colorless (as are the
3 30 10
water and salt produced in the neutralization reaction), pH is
measured to monitor the reaction. Suppose that the equivalence
Let A denote the event that a sample is from supplier 1, and let
point is reached after approximately 100 mL of an NaOH solu-
B denote the event that a sample conforms to speciications.
tion has been added (enough to react with all the acetic acid pre-
(a) Are events A and B independent?
sent) but that replicates are equally likely to indicate from 95 to
(b) Determine P B A( | ).
104 mL, measured to the nearest mL. Assume that two techni-
2-148. Redundant array of inexpensive disks (RAID) is a cians each conduct titrations independently.
technology that uses multiple hard drives to increase the speed (a) What is the probability that both technicians obtain equiva-
of data transfer and provide instant data backup. Suppose that lence at 100 mL?
the probability of any hard drive failing in a day is 0.001 and (b) What is the probability that both technicians obtain equiva-
the drive failures are independent. lence between 98 and 104 mL (inclusive)?
(a) A RAID 0 scheme uses two hard drives, each containing a (c) What is the probability that the average volume at equiva-
mirror image of the other. What is the probability of data loss? lence from the technicians is 100 mL?
Assume that data loss occurs if both drives fail within the
2-154. A credit card contains 16 digits. It also contains the
same day.
month and year of expiration. Suppose there are 1 million
(b) A RAID 1 scheme splits the data over two hard drives.
credit card holders with unique card numbers. A hacker ran-
What is the probability of data loss? Assume that data
domly selects a 16-digit credit card number.
loss occurs if at least one drive fails within the same
(a) What is the probability that it belongs to a user?
day.
(b) Suppose a hacker has a 25% chance of correctly guessing
2-149. The probability that a lab specimen contains high the year your card expires and randomly selects 1 of the 12
levels of contamination is 0.10. Five samples are checked, and months. What is the probability that the hacker correctly
the samples are independent. selects the month and year of expiration?
(a) What is the probability that none contain high levels of
2-155. Eight cavities in an injection-molding tool produce
contamination?
plastic connectors that fall into a common stream. A sample
(b) What is the probability that exactly one contains high lev-
is chosen every several minutes. Assume that the samples are
els of contamination?
independent.
(c) What is the probability that at least one contains high levels
(a) What is the probability that ive successive samples were
of contamination?
all produced in cavity 1 of the mold?
2-150. In a test of a printed circuit board using a random (b) What is the probability that ive successive samples were
test pattern, an array of 10 bits is equally likely to be 0 or 1. all produced in the same cavity of the mold?
Assume the bits are independent. (c) What is the probability that four out of ive successive sam-
(a) What is the probability that all bits are 1s? ples were produced in cavity 1 of the mold?
(b) What is the probability that all bits are 0s? 2-156. The following circuit operates if and only if there is
(c) What is the probability that exactly 5 bits are 1s and 5 bits a path of functional devices from left to right. The probability
are 0s? that each device functions is as shown. Assume that the prob-
2-151. Six tissues are extracted from an ivy plant infested by ability that a device is functional does not depend on whether
spider mites. The plant in infested in 20% of its area. Each tissue or not other devices are functional. What is the probability that
is chosen from a randomly selected area on the ivy plant. the circuit operates?
(a) What is the probability that four successive samples show
0.9 0.8 0.7
the signs of infestation?
(b) What is the probability that three out of four successive
samples show the signs of infestation?
2-152. A player of a video game is confronted with a 0.95 0.95 0.95
series of four opponents and an 80% probability of defeating
each opponent. Assume that the results from opponents are 2-157. The following circuit operates if and only if there is a path
independent (and that when the player is defeated by an oppo- of functional devices from left to right. The probability that each
nent the game ends). device functions is as shown. Assume that the probability that