Beyond mere dice: thoughts on what mathematical probability tells us about the real world

Notwithstanding critiques of listicles, this introduction to the "Probability in the Real World" project is presented here as an overview of the subject via bullet points. Nothing of this kind is ever new, but existing general discussions of probability strike me as rather nebulous philosophy or too narrowly focussed in content. So I try to give declarative statements and to cover a lot of bases. Links go to both internal and external pages. I only write pages myself when I have something to say that's not easy to find elsewhere, and here is an index to the "internal" pages written by me.

Well-known mathematical aspects of chance in everyday life

Here are a few of the many familiar ideas from
popular science style books and textbooks.

Lesser-known lessons from mathematical probability

I give more instances as mathematical curiosities.

Perception of chance in everyday life

Risk to Individuals: Perception and Reality

Risks, in the non-technical sense of dangers, is one of the main components of our everyday perception of chance. I give a lecture on 4 aspects: but I don't have anything original to say on these topics. This is a good place to remember Following the man bites dog adage, news records what is unusual, and so is unrepresentative of reality. The extent of media coverage of different risks bears little relation to their actual prevalence, but instead is driven by the psychological factors which make risks seem more threatening or less threatening than they really are. See Ropeik's book How Risky is it Really? for an interesting discussion of those factors. This also colors our judgment of long-term changes in society, which are not represented in daily headlines.

Other pages related to risks is What really has a 1 in a million chance? and Solution #76 to the Fermi Paradox.

On mathematical models

Aside from very simple "games of chance" contexts, estimating probabilities in a somewhat objective way requires a mathematical model, which (roughly speaking) involves both assumptions quantifying where chance enters, and a specification of how observable quantities arise from the underlying chance and non-random ingredients. The often-quoted line all models are wrong, but some are useful is more memorable than helpful. I prefer for reasons explained here. I like toy models -- a simplistic model which we don't pretend will give numerically accurate predictions but instead gives insight whether some mechanism might possibly explain some qualitative observation. But my discussion of use and abuse of toy models emphasizes the following point. Another under-appreciated point is that information mediated by human choice is hard to model. What do you learn from the 6 top headlines in today's news? It is not just these 6 facts, but also an almost infinite amount of implicit information about what did not happen. A twins example underscores the point: in the context of probabilities, Finally, fields such as Decision Theory and Game Theory rely on different payoffs being translated into a common "utility" number, which (outside of "only money" contexts) seems overly difficult to do in practice.

Conceptual foundations of probability

The first four points below are articulated here at greater length.

Some common misunderstandings

On paradoxes

Many of the widely discussed paradoxes strike me as too artificial to be worth discussing. Instead I do like questions we can say in words but cannot translate into precise mathematical questions .

Fantasy settings

In a toy models of a real world phenomenon, we acknowledge lack of realism of details of the model, and seek only to know if some proposed mechanism might possibly be true. There is a separate genre which (to me) is concerned with imaginary situations, and which I therefore call fantasy. Just like literary Fantasy (which in principle could be about anything but in practice fits some established subgenre) these examples tend to fit some category such as The purpose of the latter page is to outline my diagnostics for "what to dismiss as fantasy" before or during reading it.

On predicting the future

My general discussion here emphasizes a few points. In fiction one can speculate on some particular future for human society. To me it seems self-evident that non-fiction predictions or forecasts about the future should be expressed in probability terms, but this is surprisingly seldom done. And indeed some apparently-probabilistic graphics are not so -- see the second graphic here. Substantial data from recent prediction tournaments shows that, for short-term (< 1 year, say) geopolitics, some people are better than others at assessing probabilities, but it is hard to assess accuracy in absolute terms.

Slightly off topic, here is my review of a 2013 collection of 155 short essays on the theme What Should We Be Worried About? As usual these are not probability assessments, and indeed rarely even mention likely or unlikely, but as time passes one can check whether any of these worried-about futures has indeed happened.

A challenging question is: what aspects of the more longer-term future, and how far ahead, is it reasonable to try to predict via numerical probabilities? Two thoughts.
(i) It is easy to make casual assertions like "No-one predicted the end of Soviet domination of Eastern Europe in the late 1980s" but it is better to view this as a 5%-chance event that in fact happened.
(ii) Contrary to the Black Swan thesis that the modern world was shaped by dramatic unexpected events, the majority of differences between today and one or two generations ago (e.g. increase in childhood obesity, increased consumption of espresso, increased proportion of occupations requiring a College education, increased visibility of pornography, and the manifest consequences of Moore's Law) are the result of ongoing slow and steady change. And these are hard to formulate predictions for.

This leads to a rather counter-intuitive thought. Regarding the world 50 years ahead, aspects like total population or extent of climate change (often regarded as hard to predict) are in fact easier to predict than aspects like proportion of world under democratic government . Finally it is fun to describe various  Global Catastrophic Risks in class and discuss the question how much effort should we put into prevention or ameliation of any given risk?

Probability and Statistics

The phrase Probability and Statistics for an academic area has been in wide use for several generations. A modern view (see link in (i) below), as we enter the Age of Data Science, is
[Classical mathematics statistics] assumes that the data are generated by a given stochastic data model. [Machine learning] uses algorithmic models and treats the data mechanism as unknown.
So classical statistics asks whether data is consistent with a probability model, and answering that question involves the mathematics of probability. Discussing the relation further would require another web site.

A good "over the shoulder" look at how an academic statistician uses theory and data is Andrew Gelman's Statistical Modeling, Causal Inference, and Social Science blog. I myself do not engage in technical statistical analysis, because my world has been full of people who can do it better, so there is no technical statistical analysis on this site. Instead, here we seek to explain conceptual aspects of probability that can be illustrated by real data without needing any sophisticated analysis.

Having said that, here are a few comments on Statistics.
(i) Popular science books relate a perceived historical clash between frequentist and Bayesian statistics, but these are not the actual two faces of Statistics.
(ii) In accord with the saying "if all you have is a hammer, then everything looks like a nail", the widespread inappropriate use of tests of significance was one of the great scientific disasters of the 20th century, unfortunately continuing into the 21st. On the Bayesian side, the cult of informationless priors also has some misguided devotees.
(iii) At the freshman level, it has always bothered me that no-one asks the basic question when is it reasonable to regard data as a sample (an i.i.d. sample) from some unknown distribution? If I have numerical course scores (homeworks plus exams etc) for 50 students in a class, it does seem reasonable; if I have populations of the 50 States of the USA, it does not seem reasonable. But what are the criteria here?
(iv) And here is a skeptical quote from my late colleague David Freedman.

My own experience suggests that neither decision-makers nor their statisticians do in fact have prior probabilities. A large part of Bayesian statistics is about what you would do if you had a prior. For the rest, statisticians make up priors that are mathematically convenient or attractive. Once used, priors become familiar; therefore, they come to be accepted as ``natural" and are liable to be used again; such priors may eventually generate their own technical literature. Similarly, a large part of [frequentist] statistics is about what you would do if you had a model; and all of us spend enormous amounts of energy finding out what would happen if the data kept pouring in.

Probability in Science

Probability plays a role across a broad range of Science. A glimpse of this range can be seen via my map of the world of chance categories and in the Why do we care about probabilities? page. Two of my written-up Berkeley lectures do concern science: Coding and entropy and From physical randomness to the local uniformity principle. Books on my non-technical books relating to Probability list often touch upon science topics, though curiously none attempt a broad overview, even amongst those listed under "science topics". But I have not written web pages on science topics because I don't have any novel expository ideas or novel data.

On teaching Probability

Here and elsewhere I criticize the teaching of a first course in Probability, but I must confess up front that I can't do it better: my Real World course assumes students have already taken this first course.

As discussed here, my criticism is lack of real examples compounded by a plethora of manifestly unrealistic examples. In principle there is nothing wrong with illustrating math concepts via made-up stories, but in practice there is a moral hazard in implicitly teaching that it's OK to ignore realism of models, which may explain some abuse of toy models. My list of math probability predictions which are are actually verifiable is embarrassingly short.

The issue has always been "the math takes over from the concepts". These suggestions for instructors of a very elementary course should be more widely known. My Berkeley colleague Ani Adhikari has developed a Probability course Prob 140 for Data Science students, saying

Computational power in Prob 140 allows students to solve problems that are intractable by other methods. Students also explore the standard mathematical theory graphically and by simulation, and thus develop a more firm grasp of the concepts than they might by using math alone.
This should also enable more realistic examples to be studied.

Here is my brief description of Probability treatments in introductory textbooks, popular books and Wikipedia.

Fun to do in class