Imagine that there are $N$ balls in a box, among which $m$ are white and $N-m$ are black. If one blindly picks a ball in the box, the probability that the ball is white is simply:
$$p_w = \frac{m}{N}$$
Now, Imagine that $m$ is not a constant, but the outcome of some random process; as a consequence, $p_w$ becomes a random variable, whose distribution can be derived from the distribution of $m$.
The very concept of distribution of a probability seems strange to me. In a general way, is it fine to have a probability as a random variable? Can one get a grasp on what that means and what it implies?
In this precise situation, can one still define the probability to pick a white ball, or is the best one can do is to use the expected value of getting a white ball?