$\newcommand{\P}{\mathbb{P}}$Suppose $A$ and $B$ are dependent, $C$ and $B$ are dependent, i.e. $$ \P(A \cap B) = \P(A) \P(B|A) $$ with $\P(B|A) \neq \P(B)$ and $$\P(C\cap B) = \P(C)\P(B|C)$$ with $\P(B|C) \neq \P(B)$.
Can we say anything certain about the (in)dependence of $A$ and $C$?
If not, when are $A$ and $B$ (in)dependent?
Of course $A$ and $C$ are dependent in general. But they possibly are independent. Take for example two independent Bernoulli random variables $\epsilon_1$ and $\epsilon_2$ on $\{0,1\}$. Define the events $A=\{\epsilon_1=0\}$, $B=\{\epsilon_1+\epsilon_2=2\}$ and $C=\{\epsilon_2=0\}$. Then $A$ and $C$ are independent, while $A$ and $B$ as well as $B$ and $C$ are dependent.
I don't think there is a relevant characterization of the dependence between $A$ and $C$ in the general situation.
