The main problem at hand is trying to retrieve properties on the probabilities involving two random variable, knowing that their covariance is negative.
To be more precise, here is the statement I would like to be true (but maybe it isn't). Let $n$ be a natural number, and consider $X$ and $Y$ two discrete random variables taking values in $\{0, 1, ..., n\}$ such that their sum satisfies $X + Y \leq n$. If $Cov(X,Y) < 0$ (the covariance being the usual definition $Cov(X,Y) = E(XY) - E(X)E(Y)$), can I deduce that for $0 < \ell < k < n$, we have $$P(X \geq \ell ~|~ Y \leq n-k) \geq P(X \geq \ell)?$$
Intuitively, it's tempting to say yes, as the knowledge of one of the variables being smaller should increase the other one (because of the negative covariance), but I can't find a proof. Now, getting information about a lot of events just from knowledge about expectations might be asking too much...