Home > Uncategorized > Probability and rationality — trickier than you might think

Probability and rationality — trickier than you might think

from Lars Syll

The Coin-tossing Problem

My friend Bengt says that on the first day he got the following sequence of Heads and Tails when tossing a coin:

And on the second day he says that he got the following sequence:

184bic9u2w483jpgWhich day-report makes you suspicious?

Most people I ask this question says the first day-report looks suspicious.

But actually,​ both days are equally probable! Every time you toss a (fair) coin there is the same probability (50 %) of getting H or T. Both days Ben makes equally many tosses and every sequence is equally probable!

The Linda Problem

Linda is 40 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which of the following two alternatives is more probable? 

A. Linda is a bank teller.
B. Linda is a bank teller and active in the feminist movement.

‘Rationally,’ alternative B cannot be more likely than alternative A. Nonetheless Amos Tversky and Daniel Kahneman reported — ‘Judgments of and by representativeness.’ In D. Kahneman, P. Slovic & A. Tversky (Eds.), Judgment under uncertainty: Heuristics and biases.Cambridge, UK: Cambridge University Press 1982 — that more than 80 per cent​ of respondents said that it was.

Why do we make such ‘irrational’ judgments in both these cases? Tversky and Kahneman argued that in making this kind of judgment we seek the closest resemblance between causes and effects (in The Linda Problem, between Linda’s personality and her behaviour), rather than calculating a ​probability, and that this makes alternative B seem preferable. By using a heuristic called representativeness, statement B in The Linda Problem seems more ‘representative’ of Linda based on the description of her, although from a probabilistic point of view it is clearly less likely.

  1. Mark Johnson
    July 3, 2018 at 2:13 pm

    The HHHHHHHHHH is more suspicious because it could be the result of a weighted coin. Of course, if you are assuming a fair coin in the first place …

  2. July 3, 2018 at 5:36 pm

    There is a discussion on the same topic at https://larspsyll.wordpress.com/2018/07/01/probability-and-rationality-trickier-than-you-might-think/ . It would be good to some sort of defence of the psychologists, especially given their popular influence. Whilst, as a mathematician, I wouldn’t always suggest that mathematicians are always right, it seems to me that if someone like Keynes said that the sort of argument above is entirely specious, one should at least look acknowledge such reservations and point to a logical justification of Kahneman and Tversky, and not just an appeal to their authority. (Its a pity that mathematicians as such aren’t eligible for Nobel prizes, or that people don’t just accept that we know more about maths than they do, if we know little about little else.)

    The social sciences view of what is ‘rational’ seems to be a kind of populist one, but then most people would – I suspect – base their judgements on what they had been taught and the pop media. But it seems to me that mathematicians like Keynes have a point, and that some of his economic work provides some evidence that sometimes it pays to be irrational, in the sense that one should go with the logic and not the psychology.

    The Linda problem seems to me to package up more than one misconception, besides being about two topics (banking and feminism) that need careful thought before putting one’s ideas to press. If you search my blog ( djmarsay.wordpress.com) under relevant key words you’ll find comments on most of the misconceptions in the example. If you spot anything specific to Linda, other than her packaging, I’d like to know, as I am collecting the misconceptions of popular writers.

    Basically, I think, psychologists define an experiment without recognizing how it seems to the subjects, almost as if the experimenters had some sort of direct inject into the subjects’ brains, with complete mutual understanding. If only! From what they have published I have proposed a hypothesis that many subject take a much broader view of life than do the academic experimenters, and are talking a different language and even, in effect, ‘playing a different game’. Certainly, I fail to see why we should trust the interpretations of experimenters who seem unaware of even the basic logic, let alone all the potential nuances of the situations. Unless and until they can devise experiments that only have their possible interpretation, I treat their findings with a pinch of salt.

    But if you look at their papers I think they are on to something very important, even deserving of a Nobel prize. But I trust the logic over their interpretation. (Rant over!)

    • Jan Milch
      July 4, 2018 at 5:55 am

      Dave,in my view,to many mathematicians and engineers got in to social sciences bring a long their thinking and methods rigid,Well i must admit,i one to :) started there,but got into economics and sociology.Many imply same rules on the new area,and it don´t work.Humans are not machines.The subjective element is to dominant. That will not say that math,statistics not useful but it´s a tool.

      • July 4, 2018 at 9:16 am

        Jan, As a mathematician I kind of agree. It seems to me that most mathematicians have been taught the kind of mathematics that is appropriate to the ‘harder’ (i.e., easier!) sciences and engineering, and try to apply what they know to the (‘softer’) social sciences. It does not seem to me not that their mathematics as such is wrong, bad or irrelevant, but that the notion that what works in some (very successful) applications can simply be read across to other domains without ‘engaging brain’.

        I was first invited to show an interest in finance and economics when some folk were claiming that I ought to be following their lead in the use of mathematics. Even post 2008 I fear that the folly of this is not entirely evident to everyone, and it may matter. Hence my blog.

        Keynes notion of ‘pseudo-mathematics’ seems as relevant to me as ever. In health, climate change, social care, international relations, are we be influenced by ‘the real deal’ logic, maths and science, or the populist pseudo kind. This is what I would like more social scientists to really grapple with.

        (So I think I agree with what you mean, but not how you put it!)

        P.S. about to get very distracted. Hope this communicates!

  3. Jan Milch
    July 3, 2018 at 10:29 pm

    They should read this as a startingpoint. A Treatise On Probability
    by John Maynard Keynes https://archive.org/details/treatiseonprobab007528mbp

  4. Helen Sakho
    July 4, 2018 at 1:16 am

    Please see a previous comment just posted.
    Let us shed some light on the path our students are forced to take, otherwise we are mimicking the blind leading the blind, except most of us can clearly see well enough to read, write and comment?
    We need a new agenda. Pure and Simple.

    • July 4, 2018 at 9:17 am

      Or maybe we need to rediscover and reinvigorate an old agenda? But ‘not this way’!

  5. July 4, 2018 at 8:34 am

    Helen — while all on the blog are agreed on the need for change, there is very little agreement on exactly what change is needed. Everyone has their own favorite fix for the problem. This is why we are not in a position to create change. We need to generate a consensus on the direction for change. Exactly as Kuhn said, an old paradigm does not get changed by accumulation of flaws and counterexamples. Change happens when a new, alternative paradigm emerges. I dont think that there is one currently in existence — I am working on one based on Islamic Economics foundations, and others are working on other, very different, angles.

    • July 4, 2018 at 9:25 am

      Azad, you may not think that you are in a position to ‘create’ change. But, in my experience, if ever those who could promote and help shape change get serious enough again, they will be looking at blogs like this and taking courage about their own uncertainty about the dismal mainstream. Logically, no one view is going to be adequate if taken as dogma, and the only hope seems to me to be an approach informed by the broadest possible range of views (I would add, that are logically credible and ‘sustainable’ – others may differ!).

      You’d be surprised about the influence that seemingly ‘insignificant’ outsiders, particularly ‘deviants’ can have. ;-)

  6. Helen Sakho
    July 4, 2018 at 11:10 pm

    As I have officially given up, only seconds ago, I shall just reiterate a few general points again and then address specific questions asked of me.

    NO ONE on the face of this earth had a choice of which religion they were born into. EVERYONE has a choice as to what to do with themselves as a human being after a certain level of maturity and literacy.
    Are we not all HUMAN after all? Are we not all TEACHERS after all?

    Dear Asad, I have no doubts as to your beliefs, personal commitments or teaching abilities. I do want to say, here and now and for the first and last time, that I did my own thesis (one of them and decades ago) on Islamic Banking and scored very well on it. Decades later, it is mostly Muslims who are dying of starvation, poverty, famine, torture and all things related. The devout ones are already all dead! They belonged to different belief systems within the Islamic paradigm. As did all others that loved them and stood by them devoutly. All dead or dying regardless of their religions, ethnicities, backgrounds and all involuntary attributes.
    I continue to work with my students (mostly devout Muslims who have found a home in mine (Ahle Beit) and whom I had found totally illiterate, but can now understand their purpose in life and teach others (again mostly Muslims) how to stand up for their basic human rights. You are, of course, free to focus on specific groups of people and specific angles.

    So, back to the generalities:
    Can we not put entrenched positions aside and just focus on a new paradigm?
    With a bit co-operation much URGENT work could be achieved?
    I am assuming no one here is a representative of Daesh or their non-Muslim equals. And that is the only valid assumption I am sure of making.

  7. July 9, 2018 at 1:03 am

    @Dave Marsay – thanks for the encouragement.
    @Helen — I am all for it. By all means, lets do so.

  8. July 20, 2018 at 10:16 am

    Lars, I put your question to 10 random people at my athletic club. All 10 selected B. Their explanations for doing so are similar. Option B is more likely the kinds of choices Linda would make based on her history and desires for her life. She is unlikely to be satisfied with just a job as a bank teller. It seems both you and Kahneman, Slovic, and Tversky miss the boat. These 10, like most people use the information provided, along with what they know of how and why people make choices from past experiences to forecast Linda’s choices and thereby her current situation. History and models of decision making are the basis of their conclusions.

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