Question

We have a system that has 2 independent components. Both components must function in order for the system to function. The first component has 9 independent elements that each work with probability 0.92. If at least 6 of the elements are working then the first component will function. The second component has 6 independent elements that work with probability 0.85. If at least 4 of the elements are working then the second component will function.

(a)	What is the probability that the system functions?

(b)	Suppose the system is not functioning. Given that information, what is the probability that the second component is not functioning?

Answer 1

a)
P(component A works) = P(X >= 6) = (9C6 * 0.92^6 * 0.08^3) + (9C7 * 0.92^7 * 0.08^2) + (9C8 * 0.92^8 * 0.08^1) + (9C9 * 0.92^9 * 0.08^0)
P(X >= 6) = 0.0261 + 0.1285 + 0.3695 + 0.4722
P(X >= 6) = 0.9963

P(A) = 0.9963

P(component B works) = P(X >= 4) = (6C4 * 0.85^4 * 0.15^2) + (6C5 * 0.85^5 * 0.15^1) + (6C6 * 0.85^6 * 0.15^0)
P(X >= 4) = 0.1762 + 0.3993 + 0.3771
P(X >= 4) = 0.9526

P(B) = 0.9526

P(A and B) = 0.9963*0.9526 = 0.9491

P(system works) =0.9491

b)
P(A' or B') = 1 - 0.9491 = 0.0509

P(A and B' | A' or B') = 0.9963*(1-0.9491)/0.0509
= 0.9963