$(14$ points$)$ A manufacturing line contains $4$ 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 $8$ hrs$.$ Currently the manufacturing line has $2$ operators.

a$.(7$ points$)$ What is the probability of each of M machines being down at any given time?

And what are the total hours lost?

b$.(7$ points$)$ Each machine can produce $50$ units$/$hr$.$ The cost to run machines is $$40/$hr and

the operators make $$20/$hr$.$ Is it worth hiring an additional operator?