What does it mean to have a probability as random variable?

Question

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?

Answer 1

A probability that's randomly generated is a perfectly legitimate thing to have in a model. Such a model is an example of a hierarchical model: a model in which some parameters are themselves treated as random variables. Hierarchical models come up most frequently in Bayesian statistics, but they see a lot of use in frequentist statistics, too, especially in the form of so-called random effects, which are regression coefficients that are modeled as random variables.

In your example, the marginal probability of picking a white ball at any given time will be the mean of $p_w$. In Bayesian terms, if the specified distribution of $p_w$ is its prior, then the marginal probability of picking a white ball will be the posterior mean of $p_w$.

Answer 2

Probabilities as random variables come up all the time in a wide variety of contexts.

For one example, imagine you you're trying to model probability of making a claim on a third-party property damage policy in car insurance (insuring against stuff like crashing into someone else's possessions house, car, etc). Individuals will each have different probability of generating a claim (affected by how good a driver they are, how much they drive, the condition of car they have, etc), so if you take a random policy it corresponds to taking a random member of the population of insured's -- the probability of a claim a one-year insurance policy is itself a random variable from some distribution.

(I'm ignoring things like people driving other people's cars here. Still, it conveys the idea.)