Given $n$ dice, each with $k$ faces numbered $1,\dots,k$, you're allowed to ask me what the probability of some event happening is (a subset of all the possibilities and I give a number).
What is the least amount of questions you can ask in order to figure out what the probability for each die to be $i$ for all $i=1,\dots,k$.
If I further restrict myself so that each die has a rational probability for any number, can the above be improved?
I interpret the question as follows: The dice are not fair, and they may be different; we want to determine the probabilities $p_{ij}$ of all $k$ faces of all $n$ dice by asking a minimal number of questions about the probability of events that may occur when we throw all $n$ dice.
Obviously we can do with $n(k-1)$ questions by going through $k-1$ faces for each die and using the normalisation condition for the $k$-th face. In special cases we may do better, e.g. if a die is certain to always show a particular face, but in the general case, we can't do better than $n(k-1)$ questions, since the space of possible probabilities is an $n(k-1)$-dimensional manifold and each answer yields an algebraic equation in the $n(k-1)$ unknowns that reduces the dimension of the manifold by $1$.
The case of rational probabilities seems more complicated; my guess would be that this doesn't help, but I wouldn't know how to prove it.