The power failures in a certain region occur according to a Poisson process with rate λ1 = 1/5 per week. Moreover, the duration X (in hours) of a given power failure has an exponential distribution with parameter λ2 = 1/2.
Finally, we assume that the durations of the various power failures are independent random variables.
(a) What is the probability that the longest failure, among the first three power failures observed, lasts more than four hours?
(b) Suppose that there has been exactly one power failure during the first week considered. What is the probability that the failure had still not been repaired at the end of the week in question?