Are subsets of independent events independent Intuitively, if events A and B are independent, then any subset of A should also be independent of B, that is, let $C \subset A$, then is it true that $P(C \cap B) = P(C) P(B)$?
Here is my attempt to prove this statement:
We have $P(A \cap B) = P(C \cap B) + P((A \setminus C) \cap B)$ and
$P(A)P(B) = P(A \setminus C)P(B) + P(C)P(B)$.
The independence of A and B implies that $P(C \cap B) = P(C) P(B)$ if and only if $P((A \setminus C) \cap B) = P(A \setminus C)P(B)$. In words, if A and B are independent, then any subset of A is independent of B if and only if its complement in A is independent of B.
So if my original statement is wrong, can anybody give a counter example?

You can try with $C:=A\cap B$; this is a subset of $A$ and 
$$\mathbb P(C\cap B)=\mathbb P( A\cap B) =\mathbb P(A)\mathbb P(B) $$
while 
$$\mathbb P(B)\mathbb P(C)=(\mathbb P(B))^2\mathbb P(A).$$
Unless we are in a degenerated case (that is, $\mathbb P(A)=0$ or $\mathbb P(B)\in\{0,1\}$), the set $A\cap B$ is not independent of $B$. 
