Is showing that $P(A|B) \neq P(A|{B}^c)$ enough to prove that $A$ and $B$ are not independent?

Question

At least intuitively, to me this means that event B happening/not happening affects the probability of event A happening, which may seem enough to show these two events are not independent. But is this enough to prove it, or at least show it as a counterexample of A and B being independent?

Answer 1

Yes, the result you're trying to show is true. Here's a possible sketch of the argument; I've left the details as an exercise but would be happy to clarify them if you'd like.

Claim 1: If $A$ is independent of $B$, then $A$ is independent of $B^c$.

Claim 2: If $A$ is independent of $D$, then $\mathbb P(A \mid D) = \mathbb P(A)$.

If you believe both these claims (and the first one in particular warrants a proof), then it follows that if $A$ and $B$ are independent, then $\mathbb P(A \mid B) = \mathbb P(A) = \mathbb P( A \mid B^c)$. Hence, having $\mathbb P(A \mid B) \neq \mathbb P(A \mid B^c)$ would imply the dependence of $A$ and $B$.