Suppose the diagram of an electrical system is as given in the following picture:
Lets assume that the probability that any given component works is independent with probability P. meaning that not working component probability is 1-P.
I found that the probability that the system is working(signal from A <-> B) is $$F=P^2(P+P^2-P^3)(3P-3P^2+P^3)$$
im trying to find the probability that: knowing the system is working what is the probability that at least one of the components 5\7 not working.
(a) i tried doing it using $P(not(5)\cup not(7)|F)$. but im not sure if i did it correctly so ive tried doing it this way:
(b) i calculated the probabilty that 5 is not working and the system does added the probabilty that 7 is not working and the system does, added the probability that 5and7 not working but the system does.
ive got two diffent answers, im wondering if what i did on (b) is wrong and why or maybe i made i mistake when solving like (a)