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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, and plan $2$ involves providing a backup for each machine. In either case, three machines $(A,B,$ and $C)$ would be used with reliabilities equal to the original three.

a$1.$ Compute overall system reliability under Plan $1.($Round your intermediate calculations and final answer to $4$ decimal places.$)$

Reliability

a$2.$ Compute overall system reliability under Plan $2.($Round your intermediate calculations and final answer to $4$ decimal places.$)$

Reliability

a$3.$ Which plan will provide the higher reliability?