Let $\mathcal{P}_i$ be the set of probability density functions to which $f_i$ belongs, $(i=0,1)$. Furthermore assume that $L(y)=\frac{f_1(y)}{f_0(y)}$ is an increasing function for any chosen $f_1$ and $f_2$. Let the support of the densities be a compact set in reals defined by $\mathbb{K}$.
For a given threshold $\tau\in\mathbb{K}$ one can calculate the probability of false alarm and probability of miss detection as follows:
$P_F(\tau)=\int_\tau^{\infty}f_0(y)dy$
$P_M(\tau)=\int_{-\infty}^\tau f_1(y)dy$
ROC:=$(P_F(\tau),P_M(\tau))$ forms a curve in $[0,1]$ which is convex.
(ROC, for those who don't know, stands for Receiver Operator Characteristic).
Here is an example:
$f_0(y)=\frac{1}{\sqrt{2\pi\sigma_0^2}}e^{\frac{-\left(y-\mu_0\right)^2}{2\sigma_0^2}}$
$f_1(y)=\frac{1}{\sqrt{2\pi\sigma_1^2}}e^{\frac{-\left(y-\mu_1\right)^2}{2\sigma_1^2}}$
with $\sigma_0=\sigma_1=1$ and $\mu_0=0$ and $\mu_1=1$. Then we have the following figure for $(P_F(\tau),P_M(\tau))$ when $\tau$ is changed from $-\infty$ to $\infty$, ($\mathbb{K}=\mathbb{R}$).
As known and can be seen from the figure, the blue curve is convex.
For any chosen pair of densities $(f_0,f_1)\in \mathcal{P}_0\times \mathcal{P}_1$. The ROC curve (the blue one) $(P_F(\tau),P_M(\tau))$ when $\tau\in (-\infty,\infty)$ will lie in the butterfly given in the figure with red lines assuming that the point $\theta=P_F=P_M$ is common for all densities in $\mathcal{P}_0\times\mathcal{P}_1$ (in the figure $\theta \approx 0.3$)
Question:
Assume that all densities $(f_0,f_1)\in\mathcal{P}_0\times\mathcal{P}_1$ are known to have a particular $\theta$ in their ROC. In other words, let $\mathcal{P}_0\times\mathcal{P}_1$ define only the pair of densities that have $\theta$ in their ROC and furthermore let one choose any pair of density from $\mathcal{P}_0\times\mathcal{P}_1$ with equal probability.
What is the probabilty that a single point of the ROC that we obtain by this selection will lie in the green sector?
Once again the green sector is the intersection of the butterfly with the area under the line which passes through $\theta$ and $f_1/f_0$ is increasing as defined before. One can assume any $\mathbb{K}$ for example $\mathbb{K}=[0,1]$ or ($\mathbb{K}=\mathbb{R}$).