A System. The four components 1-4, in a system (shown), are operational in a tim

Question

A System. The four components 1-4, in a system (shown), are operational in a time interval Ar with the probabilities of 0.1, 0.2, 0.3, and 0.4, respectively . The components are independent. (a) Find the probability that the system is operational during the time period . 3 2, E=0.2 09 Figure 1: System components 1-4 are independent and operational with probabilities 0.1, 0.2, 0.3, and 0.4, respectively. (b) If the system was operational during the time period Ar, what is the probability that the component 3 was operational?

Explanation / Answer

(a) The system will be operational only when the the circuit is operational. Here in this circuit, component 3, component 4 and component (1 and 2) in series are parellal. So, if any one of these is operational, system will work

so P(System) = P(3) ll P(4) ll P(1) P(2)

as the system will fail only when all three circuit line will fail.

P(system) = 1 - (1 - 0.3)* ( 1 - 0.4) * ( 1 - 0.1 * 0.2) = 0.5884

(b)

system is operational so we have to find the probability that component 3 was operational.

Pr( Component 3 was operational l System is operational) = 1- Pr(Component 3 is not working and system is operational)

That means other two lines other than component 3 line is operational.

so Pr( System will work by the other two lines) = 1 - (1 - 0.4) * (1 - 0.02) = 0.412

Pr( Component 3 was operational l System is operational) = 0.3/0.5884 = 0.5099

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