1. Title
A system is composed of redundant components as shown in the figure below. If the thousand-hour reliability of each component is R, the thousand-hour reliability of the system is:
A (1-R^3)(1-R^2)
B (1-(1-R)^3)(1-(1-R)^2)
C (1-R^3)+(1-R^2)
D(1-(1-R)^3)+(1-(1-R)^2)
2. Theory
Although it is about system reliability, it is actually a question of pure mathematical probability. Basic principles:
1. If the reliability is A, the unreliability is 1-A
2. If it is two components in series, the reliability is A and B respectively. Because it is connected in series, both components are reliable, so the overall reliability is AB.
3. If two components are connected in parallel, the reliability is A and B respectively. Because of the parallel connection, the system is unreliable if the two components are unreliable, so the probability of unreliability is (1-A)(1-B), so the reliability is 1-(1-A)(1-B).
3. Problem solving
Looking at the left part of the picture, three components are connected in parallel, so the reliability is 1-(1-R)^3.
Looking at the right part of the figure, two components are connected in parallel, so the reliability is 1-(1-R)^2.
As a whole, the left and right parts are connected in series, so the overall reliability is (1-(1-R)^3)(1-(1-R)^2). That is, the answer is B.