$(10$ points$)$ A manufacturing line contains $3$ machines. $On$ average, the machines run

unattended $65\%of$ the time. The operator must perform maintenance $on$ the machines $35\%of$

the time. Each workday last $8hrs.$ Currently the manufacturing line has $1$ operator.

$a.(6$ points$)$ What $is$ the probability $ofM$ machines being down $at$ any given time? And

what are the total hours lost? Hint: $M=0,1,2,3$

$b.(4$ points$)$ Each machine can produce $50$ unit$\frac{s}{h}r.$ The cost $to$ run machines $is$ $$\frac{50}{h}r$ and

the operators make $$\frac{40}{h}r.Isit$ worth hiring $an$ additional operator? How many?