A tunneling machine bores a tunnel for a subway through rock. The reliability of this machine follows a gamma distribution with a mean of 60 days and a standard deviation of 8 days. Periodic maintenance (oil changes and lubrication) occur to ensure that at any time, the tunneling machine has a 97% reliability.

a) How often should the tunneling machine be taken out for maintenance?

b) If the tunneling machine is scheduled for maintenance and it is fully operational, the mechanic wants to skip the maintenance. If they do so, what is the probability that the tunneling machine will only function for another month?

c) A contractor purchases 3 tunneling machines at the same time. They are boring parallel tunnels for an under-the-bay artery. Workload on each machine is identical and failures can be assumed statistically independent. What is the probability that 1 machine will fail before the first maintenance?