Probability

Andrew will lead discussion on this on 14 September 2018.

The focus for the discussion will be the last part of this essay, dealing with the meaning of probability. The first two parts of the essay - about Counterfactuals and Hypotheticals - were discussed in February 2017.

Additional resources for this topic are:

I have not read all of the above, but my basic understanding is that there are two main competing interpretations of probability:

- Frequentist, which says that probability is an objective, inherent aspect of the physical world.

- Bayesian, which says that probability is subjective rather than objective (we could also say it is epistemological rather than ontological), and reflects the subject's degree of knowledge about the imaginable outcomes. Those of you who have studied probability or statistics will recognise the name Bayes, as the person who invented Bayes Theorem. And there is indeed a relationship between Bayes Theorem and the Bayesian view, as Bayes Theorem is about updating our subjective probability assessments.

Probability is arguably a concept for which we have an intuitive feel, but which we are unable to explain. We have to rely on the assumption that the person to whom we are talking has the same intuitive feel as we do. Or do we? Would anybody like to have a go at defining what is meant by saying that the probability of X occurring is p?

The notion of probability is intertwined with that of randomness. 'Random' is another term for which many feel they have an intuitive understanding, but are unable to put that understanding into words. For those interested in going down the 'randomness' rabbit hole, there's a piece I wrote here about it.

Here are some questions for consideration and discussion:

    1. Do you feel that you understand what statements about probability mean? Are you able to articulate that understanding, in a way that would get the concept across to somebody that does not feel they understand it? Note that the explanation would have to avoid using words like 'likely' or 'expected', which themselves rely on the notion of probability.
    2. What do we mean when we say something is 'expected'?
    3. What do we mean when we say two things are 'equally likely'?
    4. Do you think it is meaningful to say that, based on a sample of measurements that were taken, the mean height of men in Australia lies between say 176cm and 180cm with probability 90%? If so, what do you think it means? If not, why not? Note that such a statement is Bayesian, and my understanding is that it would be rejected as meaningless by Frequentists. I have read that Frequentists can find alternative ways about doing the same calculations (which typically involve the use of Bayes' Theorem), but I haven't got to the bottom of what they are.
    5. A surgeon tells a parent that their three-year old daughter, who is in a coma with internal abdominal bleeding following a car accident, has a 98% chance of a successful outcome of the operation, with complete recovery of health. In the light of the above material, it seems doubtful that anybody can explain what that 98% means. Yet despite the lack of any explicable meaning, the parent is so relieved that they dissolve in tears of gratitude. Why?
    6. Here are few well-known 'paradoxes' to get you thinking about just how elusive notions of probability can be:
      1. The Sleeping Beauty paradox. https://en.wikipedia.org/wiki/Sleeping_Beauty_problem. FWIW, I am a Halfer, and am absolutely not prepared to die in a ditch in defending my position from criticism by scurvy Thirders.
      2. The Two Envelope problem, in the variation thereof in which you look in the envelope you have chosen and see the amount therein, before making your probability assessments, or deciding whether to switch. People debate whether what you learn from looking in the envelope is useful new information that changes your assessment of probabilities. In addition, the envelope contains an IOU, not an amount of cash, and you are told that the lower of the two amounts is an irrational number (so it has an infinite number of non-repeating decimal places), but that only the first two decimal places are shown on the IOU. This prevents the use of strategies such as switching if the amount in the envelope is odd.
      3. Imagine a lottery in which there is one ticket for each positive integer (so that there is an infinite number of tickets), and you are randomly given a ticket. You are told that you are equally likely to be given any of the tickets. What do you think the chances are of winning the lottery? Why? Or is there something wrong with the question? If so, what is it?

The Sleeping Beauty paradox has much overlap with consideration of the Anthropic Principle, which deals with considerations of how likely we are to be in a universe whose conditions are such as can support life. This in turn relates to popular philosophical debates such as: