# How good is this bound?

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 $$\sqrt{{\bf P}(A) {\bf P}(B)} \ge \min({\bf P}(A),{\bf P}(B)) \ge {\bf P}(A \cap B)$$
• Fair. Recording another thought here, in case others find it useful: you might ask, okay what if $1 < p < \infty$, $q$ solves $1/p + 1/q = 1$? Holder's inequality also gives you the bound $\mathbf{P}(A \cap B) \leq \mathbf{P}(A)^{1/p} \mathbf{P}(B)^{1/q}$, but a similar argument above holds; this is also not useful. – Drew Brady Jan 3 at 6:47