Independence of intersections Let $(\Omega,\mathfrak{A},P)$ be a probability space and $A,B,C\in\mathfrak{A}$ some events where $A$ and $B$ are independent. I'm a bit confused now as I intuitively think that this also implies independence of $A\cap C$ and $B\cap C$. Am I right? Unfortunately, I immediately get stuck when I try to use the definition.

Two events $A$ and $B$ are independent $\iff$ $P(A \cap B) = P(A)P(B)$. This alone does not tell us anything about the third event $C$.
As a counterexample, consider flipping two coins. Call the event that the first one comes up heads $A$ (so $P(A)=.5$), the event that the second one comes up heads $B$ (so $P(B)=.5$), and the event that they both come up tails $C$ (so $P(C)=.25$). $A$ and $B$ are certainly independent (our second coin does not care how our first coin came up). But clearly $P(A \cap C)=0 \neq P(A)P(C)$ (since the first coin cannot come up both heads and tails), so $A$ and $C$ are not independent. Similarly for $B$ and $C$.
