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.

As @Michael commented, a counter-example can be posed as follows: Consider a regular die and throw it twice consecutively. Considering the events:
$ A :=\{ \text{get 2 at first} \}$, $ B :=\{ \text{get 5 at second} \}$, and $ C :=\{ \text{the sum of the two results is 7} \}$. You can see that:
$$P(A)=P(B)=P(C)=\frac{1}{6},$$
$$P(A\cap B)=P(A\cap C)=P(B\cap C) = \frac{1}{36},$$
$$P(A\cap B\cap C) = \frac{1}{36}.$$
Thus, these events, although pair-independent, they are not independent mutually:
$$P(A\cap B\cap C)\neq P(A)P(B)P(C).$$
Please, let me know if this helps or if you find that something is missing.
