Let A and B be two machines, each engaged in performing some never-ending task, with the additional feature that A is able to scan B, recognize whenever B is not performing correctly, stop B, repair B, and then start B again, and that B is able to do the same for A. Assume that A and B operate simultaneously during discrete time intervals, and that each machine is able to detect and repair a malfunction in the other machine during the single time interval in which it occurs, unless the repairing machine breaks down itself during that time interval, in which case neither will be repaired. If either machine is working correctly at time t, then the probability is p that it will break down during the interval until time t + 1; if neither machine is working correctly at time t, then it is certain that they will not be repaired at time t + 1.

(a) What is the probability that both machines will break down during the same time interval?

(b) What is the mathematically expected number of time intervals that one machine would survive alone?

(c) What is the mathematically expected number of time intervals that the two machines will survive together?