Question

A manufacturing process has stages A, B, C, D, E, that operate in a "series," which means that the process as a whole will work if and only if every stage works. Symbolically, we can indicate this as

A→B→C→D→E →Final product.

Stage A is 99.9% reliable (i.e., it will work 99.9% of the time). The other stage reliabilities are 0.959, 0.905, 0.951, and 0.963, respectively. Let W={entire process works} and A = {stage A works}, B = {stage B works}, and likewise for the other stages. Assume that each stage operates (or fails) independently of any other stage. What is P(W), the probability that the process works, to three decimal places?

Answer 1

The probability that stage A will work = 99.9/100 = 0.999 = P(A)

The probability that stage B will work = 0.959 = P(B)

The probability that stage C will work = 0.905 = P(C)

The probability that stage D will work = 0.951 = P(D)

The probability that stage E will work = 0.963 = P(E)

All the stages operate independently

Since the process involves all the stages in series, the process will work only if all the stages work

Therefore the probability that process works = P(W) = P(A).P(B).P(C).P(D).P(E) = 0.999*0.959*0.905*0.951*0.963 = 0.794