# Which math probability predictions are actually verifiable?

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

This is part of a larger program to articulate what mathematical probability says about the real world. This particular part is

• Compose a list of interesting theoretical predictions involving randomness in the real world that are actually verifiable by a junior or senior undergraduate majoring in Mathematics or Statistics, as 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.

Before saying more precisely what I mean by predictions and by actually verifiable, let me give a brief list of examples.

## Predictions that I will bet \$100 are actually verifiable

Envisaging a reader familiar with mathematical probability, I have just written brief phrases.

## Designing a probability lab project

Not everything taught in undergraduate physics can be tested in an undergraduate lab, but some parts can -- and that's why there are lab courses. And my undergraduate course is in part intended as an analgous "lab course" in probability. Not in a trite "throwing dice" sense but in a more serious "examining new data" sense.

This list of possible projects from last time the course was taught gives some idea but allows projects not so specifically focussed on verifying some quantitative prediction. For two illustations of what I am seeking, see

## The challenge: add some more predictions to the list

The challenge is for you (in the role of Instructor) to write out another "lab project" that I can add to my list. The requirements are
(i) It has to be (somewhat) different from the existing projects, and (somewhat) interesting for a student to do.
(ii) It cannot involve explicit games of chance or an explicit "IID random experiment".
(iii) The underlying theory that is being studied has to lead to some numerical prediction that can be investigated.
(iv) The Instructor has to state a general context in which the prediction is supposed to apply (rather than some highly specific example) via verbal description and /or hypothetical examples and/or actual examples studied previously.
(v) The student has to find "new data" to test the predictions. Preferably from a "new example", alternatively from "new data" for a previously studied example (e.g. in sports or the stock market). "New data" means "not previously used to test this prediction"; typically projects will involve finding data collected by other people for some other purpose. Also, typically the student will collect data from different real-world examples.

Note that for present purposes I don't care whether or not the data ultimately confirms the prediction; I write "actually verifiable" to mean "actually verifiable or falsifiable".

In particular, I am inclined to assert that there are large areas of applied probability (including those listed next) in which no numerical prediction is actually verifiable -- prove me wrong!

• Queueing theory
• Occurence of power laws
• Game theory (outside a lab setting, of course)

## Other data pages

Here are links to other pages emphasizing data -- the data presented in class and/or gathered in student projects.