Suppose we have two events $A, B \subset \Omega$ in a probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and we want to know the probability of $A \cap B$.
If $A$ and $B$ are independent, of course $\mathbf{P}(A \cap B) = \mathbf{P}(A)\mathbf{P}(B)$.
However, what can we say without independence? It seems that with Cauchy-Schwarz we have that $$ \mathbf{P}(A \cap B) = \mathbf{E} 1_A 1_B \leq \sqrt{\mathbf{P}(A) \mathbf{P}(B)}. $$ Is this a good bound? Is it useful? That's a soft question, but I guess I am wondering if it is really useful at all.
This bound is not really useful because $\min({\bf P}(A), {\bf P}(B))$ is better. Note that $$\sqrt{{\bf P}(A) {\bf P}(B)} \ge \min({\bf P}(A),{\bf P}(B)) \ge {\bf P}(A \cap B)$$