Solve these problems and upload your solution to Canvas. You can solve them $$by hand$$ or

using Excel.

Show all your work $($typed in the document, pasting a scanned image of your handwritten

solution, or by uploading an Excel file with your solution$).$ Also, provide your answer in the

provided tables.

Problem $1(12$ points$)$

A production line has three machines A$,$ B$,$ and C$,$ with reliabilities of $0\mathrm{.}90,0\mathrm{.}95,$ and $0\mathrm{.}92,$

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$.$ Compute overall system reliability under Plan $1.$

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

b$.$ Compute overall system reliability under Plan $2.$

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

c$.$ Which plan will provide the higher reliability?

$($i$)$ plan $1$

$($ii$)$ plan $2$