Consider a system consisting of four components, as pictured in the following diagram:

Components 1 and 2 form a series subsystem, as do Components 3 and 4. The two subsystems are connected in parallel. Suppose that P(1 works) = .9, P(2 works) = .9, P(3 works) = .9, and P(4 works) = .9 and that these four outcomes are independent (the four components work independently of one another).
a. The 1€“ 2 subsystem works only if both components work. What is the probability of this happening?
b. What is the probability that the 1€“ 2 subsystem doesn€™t work? that the 3€“ 4 subsystem doesn€™t work?
c. The system won€™t work if the 1€“ 2 subsystem doesn€™t work and if the 3€“ 4 subsystem also doesn€™t work. What is the probability that the system won€™t work? that it will work?
d. How would the probability of the system working change if a 5€“ 6 subsystem was added in parallel with the other two subsystems?
e. How would the probability that the system works change if there were three components in series in each of the two subsystems?

3