Reliability engineers often work with systems having components connected in parallel. In this problem, we will interpret the phrase “in parallel” as follows: The system is reliable (i.e., it is functioning) if at least one of the components is functioning. As a frame of reference, consider a two-engine aircraft. • If both engines are functioning, the aircraft is functioning (at least in lieu of non-engine related problems). • If only one engine is functioning, the aircraft still functions (although losing one engine would warrant an immediate landing). • If both engines are not functioning, the aircraft is not functioning. In this problem, we will denote by n the number of components in a parallel system.

a) Suppose n = 2. If the two components are functioning independently, each with proba- bility p, show that the system reliability r2 is given by r2 =1−(1−p)2.

b) Generalize the result in part (a) to consider a parallel system with n components (each functioning independently with probability p). That is, show that the system reliability is rn =1−(1−p)n.

c) I have a parallel system with n = 4 components (each functioning independently with probability p). How unreliable can the individual components be and still have a system with reliability r4 = 0.999?

d) So far in this problem, we have made two critical assumptions: (A1) the components function independently (A2) the components each function with the same probability p.

Give a real-life example where Assumption (A1) is likely violated. Give a real-life example where Assumption (A2) is likely violated. Do not use the plane example I used at the outset. Explain your examples sufficiently so I can understand.