Suppose we have two random variables $\xi,\eta$, which are defined on the same probability space and $\xi \sim \mathcal{U}[0,1]$. Moreover, $\forall \lambda\in[0,1]:(\eta|\xi=\lambda)\sim \mathcal{Be}(\lambda)$.
I need to find the distribution of $\eta$. In order to do so, I need to calculate $f_{\xi}(\lambda|\eta=t)$, but how does one compute it? In the question conditional probability combining discrete and continous random variables user Wio computes it using joint probability density, but is it really possible, given that one of the variables is absolutely continuous and the other is discrete?