1. About Probability

Colored Socks

You have five socks in a drawer.

Three are black and two are blue.

It is early, the room is dark, and you want a matching pair.

You could turn on the light, but then your spouse would wake up.

Three of the five socks are black. So you may think your chances of pulling out a black pair are pretty good.

You reach into the drawer and take two.

One is black. The other is blue. Bad luck, maybe.

But what were your chances in the first place?

There are only a few possibilities we need to consider.

You might draw two black socks. You might draw two blue socks. Or you might draw one of each.

But those three possibilities do not have the same weight.

There are more ways to end up with one black and one blue sock than with either matching pair.

So simply listing the possible outcomes does not yet tell us how likely they are.

How can we use what we know about the socks in the drawer to work out what weight each possibility should carry?

Probability Theory

Logic let us ask which possibilities remained compatible with fixed premises and mappings.

Probability lets us go further when several possibilities remain open and assign different weights to them. Under suitable conditions, the result can be convincingly constrained.

Where does that convincing force come from?

The previous investigation of logic began with very small formal systems. In small cases, the relevant possibilities could remain open to inspection. We could see what had been represented, which mappings had been fixed, which possibilities survived, and where a counterexample remained.

Probability offers cases of the same kind.

Dice and cards give us examples in which the relevant possibilities are easy to inspect. A die has only six faces, two throws still give a manageable number of pairs, and after drawing a card we can inspect what remains.

These examples are elementary enough that some of the structure later compressed into probability formulas stays visible.

We will also look at Bayesian updating.

Probability first lets us work forward from a model and a distribution to the probabilities of possible results. But inquiry often gives us the problem in the other direction. We observe a result and want to know how it should change the weights we assign to the possibilities that might have produced it.

Bayes' theorem gives us a simple case in which that updating can still be reconstructed from conditional probability and algebra.

More elaborate statistical methods usually rely on additional mathematical machinery that we would first have to reconstruct, even when much of it is not directly relevant to the question this essay is asking.

We will therefore follow a small path from dice to cards and then to Bayesian updating, keeping enough of the relevant possibilities and mappings visible to inspect how probabilistic constraint is constructed and how it changes when new information arrives.

As in the logic essay, this essay is not an introduction to probability theory. It will leave almost all of probability theory untouched. It does not depend on choosing one interpretation of probability over another. And although Bayesian updating is one of the cases we examine, the essay does not argue that Bayesian methods are preferable to other statistical approaches. Appendix B briefly broadens the picture with two frequentist examples, randomization and sampling, which handle evidence differently.