A parallel system functions whenever at least one of its components works. Consider a parallel system of n components, and suppose that each component works independently with probability 0.5. Find the conditional probability that component 1 works, given that the system is functioning.

P (F/W) = 0.5 / (1-0.5^N)

Step-by-step explanation:

Since each component is independent of the others then the probability that the system works:

P (W) = probability that the system works = 1 - probability that the system do not work

the system will not work only if the N components fail, then

probability that the system do not work = (1-p) ^N

where

p = probability that a component works = 0.5

thus

P (W) = 1 - (1-p) ^N = 1 - 0.5^N

then we can use the theorem of Bayes for conditional probability. Defining the event F = the component 1 works, then

P (F/W) = P (F∩W) / P (W) = P (F) / P (W) = 0.5 / (1-0.5^N)

P (F/W) = 0.5 / (1-0.5^N)

where

P (F/W) = probability that component 1 works, given that the system is functioning

P (F∩W) = probability that the component 1 works and system functions = P (F) (if the component 1 works, the system will automatically work)