Which math probability predictions are actually verifiable?

Action speaks louder than words -- but not nearly as often. Mark Twain

This whole "real world" project started around 2000 when I asked myself

After 350 years study of mathematical probability, what theoretical predictions involving randomness in the real world are actually verifiable by finding interesting recent data?
In the context of teaching a course, I meant verifiable by a junior or senior undergraduate majoring in Statistics, as part of a course project. As a non-example, statistical physics predicts that the velocities of air molecules are multivariate Normal, but I don't expect my students to be able to verify this experimentally. Similarly, statistical theory says that the effectiveness of medical treatments is better assessed via randomized controlled experiments than via anecdotal information found on the Internet; but again I don't know how students could verify this in a course project. Ideally Some examples are readily found in introductory textbooks. I had rather naively assumed that there were many more, and sought to teach a course by starting each lecture with some interesting data which motivates some less elementary theory. 18 years later I realize this was unduly optimistic.

Freshman math textbook examples

I will not discuss these except to link to some actual data examples.

The kind of examples that should be in Senior math textbooks (but rarely are)

These are the kind of topics I treat in my course, and on this site.