Fix $\epsilon, 0\leq \epsilon\leq 1/2.$ Let $Z_1,Z_2$ be zero mean, unit variance Gaussian random variables which are jointly Gaussian with $\mathbb{E}Z_1Z_2=-(1-2\epsilon)\leq 0.$

Then,

$$P(Z_1Z_2>0)\geq \epsilon.$$

This curious fact popped out of some calculations I was doing using the Central Limit Theorem. I wonder if it can be proved in some easy way by directly working with the double integral.

I also believe that the inequality is strict when $0<\epsilon<1/2.$ Can this be shown?