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Since $P(Z_1Z_2>0)=\frac12-\frac1{\pi}\arcsin(1-2\epsilon)$ hence, indeed,$P(Z_1Z_2>0)\geqslant\epsilon$, and the inequality is strict except when $\epsilon=0$, $\epsilon=\frac12$ and $\epsilon=1$.
$P(Z_1Z_2>0)=\frac12-\frac1{\pi}\arcsin(1-2\epsilon)$ hence, indeed,$P(Z_1Z_2>0)\geqslant\epsilon$ and the inequality is strict except when $\epsilon=0$ and $\epsilon=1$.
Since $P(Z_1Z_2>0)=\frac12-\frac1{\pi}\arcsin(1-2\epsilon)$, indeed $P(Z_1Z_2>0)\geqslant\epsilon$, and the inequality is strict except when $\epsilon=0$, $\epsilon=\frac12$ and $\epsilon=1$.
$P(Z_1Z_2>0)=\frac12-\frac1{\pi}\arcsin(1-2\epsilon)$ hence, indeed, $P(Z_1Z_2>0)\geqslant\epsilon$ and the inequality is strict except when $\epsilon=0$ and $\epsilon=1$.
