# Overview of Probability in the Real World project.

This web site is part of a project to articulate what mathematical probability says about the real world. The project has two faces.

1. It is intended as a complement to undergraduate mathematically-focussed courses. I teach a junior-senior "topics" course on this material. Following the link to the Fall 2011 class-by-class topics gives a bottom-up view of what I actually can do, and draft write-ups of many lectures. This page starts a top-down discussion of what I would like to be able to do.

2. The material is also intended for a general audience -- people who read "serious popular science" books. Though given my track record for actually finishing books, it may just stay forever as a web site. As such it provides a more idiosyncratic complement to two existing web sites: Understanding Uncertainty and Chance News.

This is a beta version cover page, currently used in talks to academics, urging them to consider teaching a course in this style.

## High-level organization

It took me a long time to decide on a high-level organization of this material, but currently I like the following division into 3 parts:
1. Fiction
2. Fact
3. Perception
Let me explain what I mean, in the context of practical aspects of teaching a course.

1. Saying "Hamlet is fiction" is not dissing Shakespeare. Similarly, saying

• probability models are fiction
is not dissing people like myself who study the mathematics of models; it's just a crisper version of
• As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Einstein.
• All models are wrong, but some are useful. Box and Draper.
I want students/readers to see examples of what kind of assumptions go into models, and what mathematical conclusions about the models can be obtained -- but without doing any mathematical derivations. So what this comes down to is going through some examples on a list of favorite classical applied probability models and exact formulas therein. This is easy to teach; one can spend 5 or 50 minutes talking about each model ...........

2. Turning to the second part, let me start with a narrow interpretation of "fact":

So this is a kind of "lab course" in probability. 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. Obvious examples they can verify include the birthday problem or the regression effect, so it's not hard to list 10 or 20 such -- but are there 100?

As their main requirement, my students actually do course projects -- in practice, not as sharply focussed as I would prefer -- and here are some interesting student project write-ups.

3. Though only 1/3 of the course, perception of probability is the main focus of this web site, so follow the link to get started.

## Background and goals

Reminder: This is a beta version cover page, currently used in talks to academics, urging them to consider teaching a course in this style.

Adding one word to the opening statement, my goal is to articulate critically what mathematical probability says about the real world. Here are several critiques of existing course and book material, and indications of what I seek to do instead.

Critique of introductory mathematical probability courses. A typical introductory textbook, in its introduction or back cover, makes extravagant claims about the usefulness of mathematical probability, but very little in the actual book demonstrates this usefulness -- such demonstation being implicitly postponed to future courses. What one sees in more advanced courses and research literature is "complex fiction"-- models that use technically sophisticated mathematics (compared to the 10 simple models I mention) -- but the vast majority of models are never actually checked against data. So

Critique A: most of the content of introductory mathematical probability courses serves as ``technical prerequisites for complex fiction" rather than saying something interesting about the real world.

The focus of my parts 1-3 is different; I want a course which is satisfactory as a terminal course (analogous to the Freedman et al. Statistics course), while hopefully whetting the curiousity of occasional students and motivating them to study the subject further. Closely related is

Critique B: even in the best textbooks, the majority of examples and exercises are ``just made up" -- see e.g. this list of exercises.

To phrase this more humorously, I urge instructors to

This has a conceptually easy solution: use real data. If you can't find real data on a topic, then don't teach it. Duh!

If you really care, here is some more rhetoric about how this course differs from a standard College course.

At some opposite extreme from mathematical probability, one can ask about the Big Picture -- what is the role of probability in Life, the Universe, and Everything? This is the domain of philosophers or writers of popular science style books. My overall opinion of such work is:

• Academic philosophers tend to write on narrow technical issues;
• The content of pop-sci books written by non-mathematicians tends to be skewed in one of two opposite directions; reiteration of the historical development, or wide-eyed recounting of fashionable topics emerging from research (the Wired syndrome). Those written by mathematicians tend to simply extract the more interesting parts of a college course, lightening the mathematics.
Of course there's nothing wrong with writing on some explicit specific topic within Probability, but

Critique C: Writers who claim, explicitly or implicitly, to be dealing with "probability in general" tend in fact to be working within some very narrow vision of the contexts in which Probability arises.

To demonstrate this, I need to exhibit a broader vision, and this is (under construction in) a representative list of perceived instances of chance in the real world.