Solutions - 5090-01

(1) Find the probability components 4 and 5 both succeed (denote $S_1$):

$$P[S_1]=P[W_4]\cdot P[W_5]=0.9\cdot 0.8=0.72$$

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(2) Find the probability the parallel block of 3 and $S_1$ succeeds (denote $S_2$):

$$P[S_2]=1-P[S_1^c]P[W_3^c]=1-0.28\cdot 0.6=0.832$$

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(3) Find the probability the series 2, $S_2$, 6 succeeds (denote $S_3$):

$$P[S_3]=P[W_2]P[S_2]P[W_6]=0.6\cdot 0.832\cdot 0.5=0.2496$$

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(4) Find the probability the parallel block of $S_3$ and 7 succeeds (denote $S_4$):

$$P[S_4]=1-P[S_3^c]P[W_7^c]=1-0.7504\cdot 0.3=0.77488$$

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(5) Find the probability the full system 1, $S_4$, 8 succeeds (denote $S$):

$$\begin{array}{c}P[S]=P[W_1]P[S_4]P[W_8]\\ \\ \gg \gg 0.8\cdot 0.77488\cdot 0.9\gg \gg \approx 0.5579\end{array}$$