A system is composed of four parts, J, K, L, and M. All four must function for the system to function. The four component reliabilities are .99, .98, .992, and .998. The designers are considering putting a .80 reliable backup at K. This backup will change the system reliability from ________ to ________.
A) 0.9762; 0.9605
B) 0.9605; 0.9762
C) 0.9605; some smaller value
D) exactly 0.9560
E) cannot be determined

Question

Quizplus · Accepted Answer

To calculate the system reliability with the backup at K, we need to find the probability that at least 3 out of the 5 components (K, J, L, M, and backup at K) function properly.

Without the backup:
P(system functions) = P(J functions) x P(K functions) x P(L functions) x P(M functions)
= 0.99 x 0.98 x 0.992 x 0.998
= 0.974171136

With the backup:
P(system functions) = P(K functions) x [P(J functions) x P(L functions) x P(M functions) + P(J fails) x P(L fails) x P(M fails)]
= 0.80 x [0.99 x 0.992 x 0.998 + (1-0.99) x (1-0.992) x (1-0.998)]
= 0.784495616

Therefore, the system reliability with the backup at K is 0.7845, which is less than the original reliability of 0.9742. The change in reliability is:

0.9605-0.9762 = -0.0157

However, the answer in the given options is B with the values swapped.

A System Is Composed of Four Parts, J, K, L