$6.(10$ points$)$ A manufacturing line contains $3$ machines. On average, the machines run unattended $\backslash (65\backslash \%\backslash )$ of the time. The operator must perform maintenance on the machines $\backslash (35\backslash \%\backslash )$ of the time. Each workday last $8$ hrs$.$ Currently the manufacturing line has $1$ operator.

a$.(6$ points$)$ What is the probability of $\backslash ($ M $\backslash )$ machines being down at any given time? And what are the total hours lost? Hint: $\backslash (\backslash$mathrm$\{$M$\}=0,1,2,3\backslash )$

b$.(4$ points$)$ Each machine can produce $50$ units$/$hr$.$ The cost to run machines is $\backslash (\backslash$$ $50/\backslash$mathrm$\{$hr$\}\backslash )$ and the operators make $\backslash (\backslash$$ $40/\backslash$mathrm$\{$hr$\}\backslash ).$ Is it worth hiring an additional operator? How many?