Which theory predictions are actually verifiable?

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

The data should be new in the sense of not used for this purpose before.
Simulations of a model are not data, but can be used to see what a model predicts, to compare with real-world data.
The topic should be (at least somewhat) interesting "outside the classroom". The sum of 100 rolls of a die is not interesting in that way.

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 birthday problem prediction, which can be nicely illustrated by baseball teams' active rosters.
The Poisson distribution approximation for counts of rare occurrences, illustrated by daily murders in London or World Cup goals by minute.
Accuracy of opinion polls, illustrated by Field California Poll records.
The regression effect illustrated by next year's performance by this year's best and worst teams.
Benford's law. See this collection of data sets.
Textbooks are full of "games of chance" calculations for poker, blackjack, craps, roulette, lotteries etc. Because the accuracy of the models here is uncontroversial, I have never been motivated to compare theory with data, but you could try with Powerball statistics, for instance.
The Bayes calculation of probability of a false positive is usually given with hypothetical figures, but it is not hard to find real data.

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.

Intra-day stock prices can be modeled quite well as random walk/Brownian motion; so one can check non-intuitive theory predictions such as the three arc sine laws.
Play conservatively when ahead, play boldly when behind, illustrated by NFL interceptions.
Theory says that prediction market prices should fluctuate as martingales, and one can check this via counting upcrossings.
For the basic textbook stochastic process topics (Markov chains, branching processes, etc) it's hard to find relevant real world data. The circle of ideas surrounding ratings in (team or individual) sports includes the Bradley-Terry model (unchanging strengths) and the Elo scheme (tracking changing strengths). These are probability models which can be compared with extensive data. See my survey article.
Shannon's theory of entropy as data compression rate applies to real language: illustrated by compressing first and second halves of Don Quixote. One could also check this for sorting algorithms like Quicksort.
It took me a long time to find interesting new data related to Game Theory, but there is an actual online game that one can observe and compare players' behavior to theoretical Nash equilibria.