Sheldon Ross 10: Exercise 3.74

Question: There are five components. The components act independently, with component i working with probability pi , i = 1, 2, 3, 4, 5. These components form a system as shown in Figure below. The system is said to work if a signal originating at the left end of the diagram can reach the right end, where it can pass through a component only if that component is working. (For instance, if components 1 and 4 both work, then the system also works.) What is the probability that the system works?


Analytical Solution

We will set up the initial equation and then proceed to expand it.

P{ Systemworks } = ( C1 && ( C4 || ( C3 && C5 ) ) ) ||  ( C2 && ( C5 || ( C3 && C4 ) ) )

P{ Systemworks } = A + B – A*B

where

  • A = ( C1 && ( C4 || ( C3 && C5 ) ) ) = ( C1 && ( C4 || ( pp5 ) ) ) = ( C1 && ( p4 + pp– p4 pp5 ) ) ) = p1(p4 + pp5 – p4 pp5)
  • B = ( C2 && ( C5 || ( C3 && C4 ) ) ) = ( C2 && ( C5 || ( pp4 ) ) ) = ( C2 && ( p5 + pp– p5 pp4 ) ) ) = p1(p4 + pp5 – p4 pp5)

Simulation Solution

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Code

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End of the post


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