Preventative maintenance tests. The optimal scheduling of preventative maintenance tests of some (but not all) of n independently operating components was developed in Reliability Engineering and System Safety (Jan. 2006). The time (in hours) between failures of a component was approximated by an exponentially distributed random variable with mean u.

a. Suppose u = 1,000 hours. Find the probability that the time between component failures ranges between 1,200 and 1,500 hours.

b. Again, assume u = 1,000 hours. Find the probability that the time between component failures is at least 1,200 hours.

c. Given that the time between failures is at least 1,200 hours, what is the probability that the time between failures is less than 1,500 hours?