So, I have no idea if I'm doing this correctly. I think I am, but I haven't received much help from my teacher and the TA is very, very sick.
The question is: there's a parallel system that is functional whenever at least one component is working. Each component independently works with probability 1/2. Find the conditional probability that component 1 works given a functioning system.
Conditional Probability formula: $P(E\mid F) = P(E\ and\ F) / P(F)$
P(E) = chance of component 1 working: 1/2
P(F) = functioning system = 1 – P(system isn't functioning) = 1 – (.5)^n, where n is the amount of components the system has.
$P(E\ and\ F)/P(F) = (1/2)(1 – (.5)^n) / (1-(.5)^n) = 1/2$
Is this correct, or am I missing something important?
Best Answer
The error in your computation comes from not calculating $\Pr[E \cap F]$ correctly. Events $E$ and $F$ are not independent. If $E$ occurs, then $F$ also occurs, because if component 1 is working, then the system is functioning. So $$\Pr[E \cap F] = \Pr[E] = 1/2.$$
If there are $n$ components in the system, then you have correctly computed $$\Pr[F] = 1 - 2^{-n}.$$ The rest should now be straightforward.