First, apologies for the image, I couldn't find a good way to explain without including it.
Number on each component represents the probability this component works
Assuming all its components are independent, what is the probability that the circuit doesn't work?
Note: for the circuit to work we need at least one line from left to right with all components working
I am posting this for two reasons:
- I think my solution is long and I might be potentially missing something to make things easier
- I got stuck somewhere while solving
I tried doing $1-P(W)$
Let $W$ be the event that the circuit works. For this, we always need $A$ and work, and then either $B$ OR $C$ AND $D$ OR $E$ AND $F$, mathematically this can be written $P(A \cap (B \cup (C\cap D) \cup (E \cap F)))$, and I am not sure how I could proceed with it, the $\cap$ is easy to do since these are independent events I can just multiply them but I don't know how to go ahead with all this $\cup$, I thought about the general formula of $P(A\cap B \cap C=P(A)+P(B)+P(C)-P(A\cap B) \dots $ but I am not quite sure of it
I also tried doing $P(W')$ directly but it seemed longer.
Any help would be appreciated.