I am currently solving the following problem about conditional probability:
"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 $\frac{1}{2}$. Find the conditional probability that component 1 works given that the system is functioning."
I think that I do have the answer to this problem; however, since textbook does not contain answer to this one, I am sharing it with the community to poke holes in my logic, if any.
Best Answer
Let's call event "whole parallel system works" as $W$ and event "first component works" as $W_1$. Our task is to find $P(W_1|W)$. Let's use conditional probability definition to expand it: $$P(W_1|W) = \frac {P(W_1 \cap W)}{P(W)} = \frac {P(W | W_1)\cdot P(W_1)}{P(W)}$$ Now, $P(W_1) = \frac{1}{2}$. $P(W) = 1 - P(\bar W) = 1 - (\frac{1}{2})^n = 1 - \frac{1}{2^n}$. And $P(W | W_1) = 1$, since the whole system is active if first component is active. As the result, we have: $$P(W_1|W) = \frac{\frac{1}{2}}{1 - \frac {1}{2^n}} = \frac{\frac{1}{2}}{ \frac {2^n-1}{2^n}} = \frac{2^{n-1}}{2^n-1}$$