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 $,$ and $C)$ would be used with reliabilities equal to the original three. Plan $1$ and Plan $2$ reliability are $0\mathrm{.}9865$ and $0\mathrm{.}9934.$

$1-$ Assume that the single switch used in plan $1$ is $98$ percent reliable, while the reliabilities of the machines remain the same. Recalculate the reliability of plan #$1.$

$($Round your intermediate calculations and final answer to $4$ decimal places.$)$

$2-$ Compare the reliability of this plan with the given reliability of plan $1.$ How much did the reliability of plan $1$ decrease as a result of a $98$ percent reliable switch?

$($Round your intermediate calculations and final answer to $4$ decimal places.$)$

$3-$ Assume that the three switches used in plan $2$ are all $98$ percent reliable, while the reliabilities of the machines remain the same. Recalculate the reliability of plan

#$2.$

$($Round your intermediate calculations and final

answer to $4$ decimal places.$)$

$4-$ Compare the reliability of this plan with the given reliability of plan $2.$ How much did the reliability of plan #$2$ decrease?

$($Round your intermediate calculations and final answer to $4$ decimal places.$)$