Reallife systems often are composed of several components Fo

Real-life systems often are composed of several components. For example, a system may consist of two components that are connected in parallel as shown in Figure 1.28. When the system\'s components are connected in parallel, the system works if at least one of the components is functional. The components might also be connected in series as shown in Figure 1.28. When the system\'s components are connected in series, the system works if all of the components are functional. s components are co 2 2 Fig.1.28 - In left figure, Components C1 and C2 are connected in parallel. The system is functional if at least one of the C1 and C2 is functional. In right figure, Components C1 and C2 are connected in series. The svstem is functional onlv if both C and C are functional For each of the following systems, find the probability that the system is functional. Assume that component k is functional with probability Pk independent of other components. C2 3 Cs 3 C2 2 C3 3 C1 C2 3

Solution

system will work in parallel if at least 1 is working and in series all are working

a) here the system is in series so it will work only if the components will work all

probability of 1 component working = Pk

therefore the probability that the system will work = Pk^3

b) here all are in parallel so the maximum probability will depend on any one

if c1 works or c2 works or c3 works hence total probability = 3Pk

c) here c1 and c2 are in parallel

there fore it will work if c1 and c3 works or c2 and c3 works

hence the probability = 2Pk^2

d) here c1 and c2 is in parallel with c3 so the probability =

Pk^2+Pk

e) here it will depend on c1,c2,c5 or c3,c4,c5

hence the total probability = 2Pk^3