Suppose $X$ and $Y$ are symmetric mean zero random varables. Assume that cov$(X,Y)\geq 0$.
Does this imply that Pr$(X\geq 0|Y\geq 0)\geq 0.5$?
Intuitively, this makes sense to me since higher values of $Y$ correspond to higher values of $X$.
Edit to reflect a suggestion in the comments Edit: probably a replication https://math.stackexchange.com/a/2373957/16397,
No, it doesn't. E.g., let the distribution of $(X,Y)$ be given by $$\begin{align}\mathbf P(X=1,Y=-1)=\mathbf P(X=-1,Y=1)&=14/30,\\ \mathbf P(X=10^4,Y=10^4)=\mathbf P(X=-10^4,Y=-10^4)&=1/30.\end{align}$$ Then $$\mathbf{Cov}(X,Y)=2{14\over 30}(-1)+2{1\over 30}(10^8)>0,$$ but $$\mathbf P(X\ge 0\mid Y\ge 0)={\mathbf P(X\ge 0, Y\ge 0)\over\mathbf P(Y\ge 0)}={1/30\over 1/2} = 1/15<1/2.$$
(This example distribution is borrowed from https://math.stackexchange.com/a/2373957/16397, which treats a similar question.)