a.
Compute the probability of 1-2 subsystem works only if both components work.
a.
Answer to Problem 51E
The probability of 1-2 subsystem works only if both components work is 0.81.
Explanation of Solution
Calculation:
The given system consists of four components and all of the four components work independently.
Multiplication Rule:
If two Events A and B are independent then
Then, the required probability is obtained by the following formula:
It is given that
Thus, the probability of 1-2 subsystem works only if both components work is 0.81.
b.
Obtain the probability of 1-2 subsystem does not work.
Find the probability of 3-4 subsystem does not work.
b.
Answer to Problem 51E
The probability of 1-2 subsystem does not work is 0.19.
The probability of 3-4 subsystem does not work is 0.19.
Explanation of Solution
Calculation:
From Part (a), the probability of 1-2 subsystem works only if both components work is 0.81.
Then, the probability of 1-2 subsystem does not work is obtained by the following formula:
Thus, the probability of 1-2 subsystem does not work is 0.19.
The probability of 3-4 subsystem works is obtained by the following formula:
Then, the probability of 3-4 subsystem does not work is obtained by the following formula:
Thus, the probability of 3-4 subsystem does not work is 0.19.
c.
Compute the probability that subsystem would not work if the 1-2 subsystem does not work and if 3-4 subsystem also does not work.
Compute the probability that subsystem will work.
c.
Answer to Problem 51E
The probability that subsystem would not work is 0.0361.
The probability of subsystem will work is 0.9639.
Explanation of Solution
Calculation:
From Part (b), the probability of 1-2 subsystem does not work is 0.19 and the probability of 3-4 subsystem does not work is 0.19.
The probability of 3-4 subsystem works is obtained by the following formula:
Thus, the probability that subsystem would not work is 0.0361.
Then, the probability of subsystem will work is obtained by the following formula:
Thus, the probability of subsystem will work is 0.9639.
d.
Explain how would be the probability of the system working change if a 5-6 subsystem were added in parallel with the other two subsystems.
d.
Explanation of Solution
Calculation:
The given system consists of four components and all of the four components work independently.
Multiplication Rule:
If two Events A and B are independent then
From Part (b), the probability of 1-2 subsystem does not work is 0.19 and the probability of 3-4 subsystem does not work is 0.19.
It is given that 5-6 subsystem is also added in parallel, then the probability of 5-6 subsystem works only if both components work is 0.81.
Then, the probability of 5-6 subsystem does not work is obtained by the following formula:
Thus, the probability of 5-6 subsystem does not work is 0.19.
The probability of subsystem would not work is obtained by the following formula:
Thus, the probability that subsystem would not work is 0.006859.
Then, the probability of subsystem will work is obtained by the following formula:
Thus, the probability of subsystem will work is 0.993141.
e.
Describe how would be the probability that the system works change if there were three components in series in each of the two subsystems.
e.
Explanation of Solution
Calculation:
The probability that one particular subsystem will work is obtained by the following formula:
Thus, the probability that one particular subsystem will work is 0.729.
Then, the probability that the subsystem will not work is as follows:
Then, the probability that the subsystem will not work is 0.271.
Then, the probability that neither of the two subsystems work is as follows:
Thus, the probability that neither of the two subsystems work is 0.073441.
Therefore, the probability that the system works is as follows:
Thus, the probability that the system works is 0.926559.
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Chapter 6 Solutions
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