Series and parallel systems. Consider the two systems shown in the schematic below. System A operates properly only if all three components operate properly. (The three components are said to operate in series.) The probability of failure for system A components 1, 2, and 3 is .1, .08, and .13, respectively. Assume that the components operate independently of each other.

System B comprises two subsystems said to operate in parallel. Each subsystem has two components that operate in series. System B will operate properly as long as at least one of the subsystems functions properly. The probability of failure for each component in the system is .2.
Assume that the components operate independently of each other.

a. Find the probability that System A operates properly.

b. What is the probability that at least one of the components in System A will fail and therefore that the system will fail?

c. Find the probability that System B operates properly.

d. Find the probability that exactly one subsystem in System B fails.

e. Find the probability that System B fails to operate properly.

f. How many parallel subsystems like the two shown would be required to guarantee that the system would operate properly at least 99% of the time?