## The nature of probability

Recently I was involved in a Twitter conversation about quantum mechanics, during which I claimed that the nature of probability has been completely solved. I was going to write a blog post about quantum theory, but my thoughts on it aren’t mature enough so I decided to focus on the completely solved nature of probability.

Probabilities are mathematical objects that obey certain rules, such as the sum and product rule. It is possible to study probability in an abstract pure mathematics type way, where you just derive consequences of axioms and that’s it.
However, when we apply probability theory to the world, we have to decide what it is that the probabilities in our equations are modelling, and whether it’s valid to do that. In this post I’ll go over the two main applications of probability theory — proportions and plausibilities.

The first main application of probability to the real world is to proportions or frequencies [e.g. x % of ys are z, the kind of statement usually associated with statisticians]. The rules of probability theory relate some proportions to others. For example, consider the proportion of people who have ten fingers (call this property $A$), and the proportion of people who have black hair (call this property $B$). These satisfy the product rule: $Proportion(A, B) = Proportion(A)Proportion(B|A)$: the proportion of people with both ten fingers and black hair equals the proportion with ten fingers, multiplied by the proportion of the ten-fingered subset that have black hair.

The other important application of probability theory is to reasoning in the presence of uncertainty. Consider the propositions $A$: Tony Abbott will be PM of Australia on January 1st, 2016, and $B$: I ate pizza on the 25th of February, 2015 (New Zealand time). The plausibilities of these propositions satisfy the product rule: $Plausibility(A, B) = Plausibility(A)Plausibility(B|A)$. That is, how plausible you find the joint proposition Tony Abbott will be PM of Australia on January 1st, 2016 and Brendon ate pizza on February 25th is (or should be) equal to how plausible you find the Abbott proposition to be, multiplied by how plausible the pizza proposition would be if you learned that the Abbott proposition was in fact true. It may not be obvious why plausibilities should satisfy the equations of probability theory, but there are many compelling arguments in favour of this (e.g. the Cox/Jaynes argument and variations thereof, the dutch book argument, etc).

Both proportions (which are facts in the world) and plausibilities (which are attitudes existing in minds) can be modelled completely legitimately using probability. Incidentally, the problem with frequentism is not acceptance of the first application, but the rejection of the second (which leads one to try answering questions of plausibility with methods that contradict probability theory).

Confusion can enter when a problem contains both proportions and plausibilities: you might want to know the plausibility of a proposition about the value of a proportion! Unfortunately most basic probability examples (coins, dice, cards, etc) have this feature.
For example, the plausibility that my next coin flip will result in heads is 50%, and if I flip it lots of times there is a very high plausibility that the proportion of heads in my sequence of results will be close to 50%.

That’s why I recently decided that I don’t think we should use coin, dice and card examples in teaching. If I were teaching an intro probability course I would try to make all of my examples so obviously about either proportions or plausibilities. The problems containing both features are advanced and should be left to later.