3.37. Crosslinked System 1. Each of the five components in a crosslinked system

Question

3.37. Crosslinked System 1. Each of the five components in a crosslinked system (shown in Fig. 3.22) is operational in a time interval [0, T] with the probability of 0.6. The components are independent. Let E denotes the event that ith component is operational at time T and Ei^c that it is not. Denote by A the event that the system is operational at time T. Fig. 3.22 Crosslinked system for Exercise 3.37. (a) Find the probabilities of events H1 = E2^c E3^c, H2 = E2^cE3, H3 = E2E3^c, and H4 = E2E3. Do these four probabilities sum up to 1? (b) What is the probability of the system being operational if H1 is true, that is, what is P(A|H1)? Find also P(A|H2), P(A|H3) and P(A|H4). (c) Using results in (a) and (b), find P(A).

Explanation / Answer

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a)

H1 = (1-0.6) ( 1-0.6 ) = 0.16

H2 = (1-0.6) (0.6)= 0.24

H3= (0.6) ( 1-0.6) = 0.24

H4 = (0.6) ( 0.6) = 0.36

0.36+0.16+0.24+0.24 =1

these 4 probabilities sum 1

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