What distributions might describe the percentage of a population with a trait across groups?

Suppose I have a large number of urns, each with a different ratio $r$ of white and black balls. Some urns may be full of white balls or full of black balls. What kinds of distributions or processes could I use to model the percentage of white balls across the urns? So I'm looking for a distribution with support $[0,1]$ that does not necessarily go to zero at the ends.

I initially thought that a beta distribution would make sense, but that assumes that the balls are coming from some independent process, and you typically get $p(0) = p(1) = 0$. In many applications, similar people tend to stick together, so you get higher probabilities for the ends.

Possible examples include:

1. Percentage of a country's population that follows Christianity (there are many countries that are 100% or 0% Christian).

2. Percentage of products made by companies that are of a particular type (there are many companies that sell exclusively clothes and many that sell exclusively non-clothes, and some that sell both).

3. Percentage of a minority group in a neighborhood.