I have this chain.
$A,B,C,D,E $ are switches. They are independent from each other. They can be switched on with probability $p$ or switched off with probability $1-p$. We need to find probability that when current get into the chain it will get out of the chain.
So I was thinking in to ways. Not sure which one is correct if it is so.
First way. We have 4 possibilities to get out from the chain: $AB, AED, CD, CEB$ $$p^2+p^3+p^2+p^2=2p^2(1+p)$$
Second way. $$\mathbb{P}\big( (A\cap B)\cup (A\cap E\cap D) \cup (C\cap D) \cup(C \cap E \cap B) \big)$$ but here I have another problem. I know that $\mathbb{P}(X \cup Y)=\mathbb{P}(X)+\mathbb{P}(Y)-\mathbb{P}(X\cap Y)$ but when I try to apply I always loose something and cannot find right probability....
So which way is correct?