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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.

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This bound is not really useful because $\min({\bf P}(A), {\bf P}(B))$ is better. Note that $$\sqrt{{\bf P}(A) {\bf P}(B)} \ge \min({\bf P}(A),{\bf P}(B)) \ge {\bf P}(A \cap B)$$

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  • $\begingroup$ 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. $\endgroup$ – Drew Brady Jan 3 at 6:47

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