A major city transports water from its storage reservoir to the city via three large tunnels. During an arbitrary summer week there is a probability q that the reservoir level will be low. Owing to the occasional call to repair a tunnel or its control valves, etc., there are probabilities pi (i = 1, 2, 3) that tunnel i will be out of service during any particular week. These calls to repair particular tunnels are independent of each other and of the reservoir level. The "safety performance" of the system (in terms of its potential ability to meet heavy emergency fire demands) in any week will be satisfactory if the reservoir level is high and if all tunnels are functioning; the performance will be poor if more than one tunnel is out of service or if the reservoir is low and any tunnel is out of service; the performance will be marginal otherwise.
(a) Define the events of interest. In particular, what events are associated with marginal performance? (b) What is the probability that exactly one tunnel fails?
(c) What is the probability of marginal performance?
(d) What is the probability that any particular week of marginal performance will be caused by a low reservoir level rather than by a tunnel being out of service?