Page 74 - Applied statistics and probability for engineers

52 Chapter 2/Probability

(
(
B )
( ⎡
′
′
P T or B) = − P T or B) ⎤ = − P T and ′
1
1
⎦
⎣
A simple formula for the solution can be derived from the complements T′ and B′. From the independence assumption,
(
2
P T′ and B′) = ( ′) − . ) = .05 2
P T P B′ ( ) = (1
5
0
0
9
so
(
P T or B) = − .05 2 = .9975
0
0
1
Practical Interpretation: Notice that the probability that the circuit operates is larger than the probability that either
device is functional. This is an advantage of a parallel architecture. A disadvantage is that multiple devices are needed.
Example 2-35 Advanced Circuit The following circuit operates only if there is a path of functional devices
from left to right. The probability that each device functions is shown on the graph. Assume that
devices fail independently. What is the probability that the circuit operates?
0.9
0.95
a 0.9 0.99 b
0.95
0.9
The solution can be obtained from a partition of the graph into three columns. Let L denote the event that there is a path
of functional devices only through the three units on the left. From the independence and based upon the previous example,
0
P L ( ) = − .1 3
1
Similarly, let M denote the event that there is a path of functional devices only through the two units in the middle. Then,
0
1
P M ( ) = − .05 2
The probability that there is a path of functional devices only through the one unit on the right is simply the probability
that the device functions, namely, 0.99. Therefore, with the independence assumption used again, the solution is
− (
1 0 1 )( 1 0 05 )( 0 99) = 0 987
−
.
.
.
.
2
3
Exercises FOR SECTION 2-6
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
| (
(
0
0
2-142. If P A B) = .4 ,P B) = .8 , and P A ( ) = .5 , are 2-146. Disks of polycarbonate plastic from a supplier are
0
the events A and B independent? analyzed for scratch and shock resistance. The results from 100
| (
(
2-143. If P A B) = .3 ,P B) = .8 , and P A ( ) = .3 , are disks are summarized as follows:
0
0
0
the events B and the complement of A independent?
(
2-144. If P A ( ) = .2 ,P B) = .2 , and A and B are mutu- Shock Resistance
0
0
ally exclusive, are they independent? High Low
2-145. A batch of 500 containers of frozen orange juice con-
tains 5 that are defective. Two are selected, at random, without Scratch High 70 9
replacement, from the batch. Let A and B denote the events that Resistance Low 16 5
the irst and second containers selected are defective, respectively.

(a) Are A and B independent events? Let A denote the event that a disk has high shock resistance,
(b) If the sampling were done with replacement, would A and and let B denote the event that a disk has high scratch resist-
B be independent? ance. Are events A and B independent?