Question

2, (20%) A particular operation has six components. Each component has a failure probability q, independent of any other component. The operation is successful if both of the following conditions hold: Component 1, 2, and 3 all work, or component 4 works. Either component 5 or component 6 works. The block diagram of this operation is shown below. b. (10%) Suppose we can replace one of the components with a more reliable component that has failure probability of q/2 Which component should we replace, and why? (Intuitively argue which component we should replace.)

Answer 1

I am using & to denote the intersection of events(). For independent events, A1, A2, ...,An: P(A1 & A2 & ... & An) =P(A1)*P(A2)*...*P(An).

Since we need all 1, 2 and 3 to be working for any of them tobe of any use, let's consider 1,2 and 3 combined as a singlecomponent I.

P(I not working) = 1 - P(I working)

                         = 1 - P(1 working & 2 working & 3 working)

                         = 1 - P(1 working) * P(2 working) * P(3 working)   [byindep]

                         = 1- (1-q)³

Now, for the overall system to work we need atleast oneof I and 4 to be working and atleast one of 5 and 6 to beworking.

P(atleast one of I and 4 working) = 1- P(I not working & 4not working)

                                                   =1 - P(I not working) * P(4 not working) [by indep]

                                                   =1 - [1-(1-q)³]* q

P(atleast one of 5 and 6 working) = 1 - P(5 not working &6 not working)

                                                   = 1 - P(5 not working) * P(6 not working) [by indep]

                                                   = 1 - q²

P({atleast one of I and 4 working} & {atleastone of 5 and6 working})

                    =P(atleast one of I and 4 working) * P(atleast one of 5 and 6working) [by indep]

                    =  {1- [1-(1-q)³]* q} * {1-q²}