In Probabilistic Reasoning For Intelligent Systems by Judea Pearl, in chapter 2, the three prisoner paradox was handled.
One scenario was raised where $P(I_{B}^{'}|G_A)=q$ is assumed; here $G_A$ is the event of prisoner $A$ (the one that asks the guard) being convicted and $I_B^{'}$ is the event of the guard reporting that prisoner $B$ will be released. Here we consider the guard's bias towards reporting $B$'s release to be an unknown ($q$).
$P(G_A|I_B^{'})$ = $\frac {q} {1+q}$.
We can even assume a probability density function $f(q)$ for the random variable $q$ that varies between $0$ to $1$, in which case -
$P(G_A|I_B^{'})$
= $\int_{0}^{1} {\frac {q} {1+q} f(q|I_B^{'}) dq}$
= $\frac {\int_0^1 {q f(q) dq}} {1 + \int_0^1 {q f(q) dq}}$
I do not understand the transformation from second to third line here.
Note - $f(q)$, being a probability density function for $q$ varying between $0$ - $1$, I assume $\int_{0}^{1} {f(q)} dq$ = $1$.