A company has an electric network with three vertices i, j and k. One line goes from i to j, which we will denote as "a" and denote "A" if the event works. There is a line b from j to k and another line c from j to k (there are no other lines). Denote B if b works and denote C if the event c works. Denote E as the event that electricity can flow from i to k. Imagine a powerplant in i and an electrical company in k. A and B are not independent, instead:
$$P(A) = 0.9, P(B) = 0.8, P(A \cap B) = 0.75$$
we guess $P(C) = 0.5$ and that any event related to a and b is independent of C. Calculate the probability of E.
My attempt at a solution:
I know that $A \cup B, A \cap B, A, B,A - B, B-A$ are all independent of C. I also know in order for A and B to be independent then $(P\cap B) = P(A)P(B)$, which is not the case here and already stated in the question. Knowing these then is $$P(E) = P((A \cap B) \cup C) = P(A \cap B) +P(C) - P(A\cap B \cap C) = [(.9)(.8) + (.5) - [(.9)(.8)(.5)]]= .86$$
Is this correct?