A production line has three machines A$,$ B$,$ and C$,$ with reliabilities of $0\mathrm{.}99,0\mathrm{.}96,$ and $0\mathrm{.}93,$

respectively. The machines are arranged so that if one breaks down, the others must shut down.

Engineers are weighing two alternative designs for increasing the line$$s reliability. Plan $1$ involves

adding an identical backup line $($i$.$e$.,$ a series backup$),$ and Plan $2$ involves providing a backup for

each machine $($i$.$e$.,$ a parallel backup$).$ In either case, three additional machines$($A$,$ B$,$ and C$)$ would

be used with reliabilities equal to the original three.

a$.$ Which plan will provide higher reliability?

b$.$ Explain why the two reliabilities are not the same.

c$.$ What other factors might enter into the decision of which plan to adopt?

d$.$ Assume that a single switch is used in Plan $1$ to transfer production to the backup line if the

first line fails, and this switch is $98$ percent reliable, while reliabilities of the machines remain

the same. Recalculate the reliability of Plan $1.$ Compare this reliability with the reliability of

Plan $1$ calculated in solving the original problem. How much did reliability of Plan $1$ decrease

as a result of a $98$ percent reliable switch?

e$.$ Assume that three switches are used in Plan $2$ to transfer production to the backup machines if

the original machines fail, and these switches are all $98$ percent reliable, while reliabilities of

the machines remain the same. Recalculate the reliability of Plan $2.$ Compare the reliability of

this plan with the reliability of Plan $2$ calculated in solving the original problem. How much

did reliability of Plan $2$ decrease?