I have a question, which is relate to the following topic.
Assuming the following circuit, the probability that the entire system works is given by: $$P[X]\cup P[Y] = P[X] + P[Y] - P[X\cap Y],$$
given P[X] is defined by the branch AB and P[Y] by CDE. Now, what is the probability that A is not working given the system works?
I would write it as
$$P[A'|Z] = \frac{P[A'\cap Z]}{P[Z]},$$ where $P[Z]=P[X]\cup P[Y].$ Intuitively, one could consider $P[A']\cap P[Z] = P[A']P[Z]$. However, this only holds when $A'$ and $Z$ are independent. Since the system works independently from branch $AB$ when $C$, $D$ and $E$ are working, one could use $P[A'\cap (C\cap D \cap E)] = P[A']P[C]P[D]P[E]$. Finally, I reach my question. Why should one still use $P[Z]$ in the denominator, i.e., $$P[A'|Z] = \frac{P[A'\cap (C\cap D \cap E)]}{P[Z]}\quad (1)$$ rather than $$P[A'|C\cap D \cap E] = \frac{P[A'\cap (C\cap D \cap E)]}{P[C\cap D \cap E]}\quad (2)?$$
Equation (1) must be used in order to reach the same answer as given in the book. However, it bugs me that the conditional probability equation requires an "adaptation".
The question is to calculate the probability that A is not working knowing the system works so $$P[A'|Z] = \frac{P[A'\cap (C\cap D \cap E)]}{P[Z]}\quad (1)$$
as you mentioned it.
The second equation you wrote is literally what is the probability that A is not functioning knowing the bottom branch is working. You would agree that the bottom branch working or not has nothing to do with A working or not, hence those two events are independent and you get simply:
$$P[A'|\cap (C\cap D \cap E)] = P[A'] * P[C\cap D \cap E]\quad (2)$$