If $X$ and $Y$ are (not necessarily independent) random variables taking values in $\Omega=\{1,\ldots,n\}$. then:
$\sum_{i=1}^nP(X=i,Y=i)\leq1-\frac12\sum_{i=1}^n\mid P(X=i)-P(Y=i)\mid$
I am only 99.9% sure this inequality is true. I hope someone can prove it. Thanks in advance!