Question

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 .

Answer 1

Answer:

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)