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

Probability and rationality — trickier than you may think

from Lars Syll

The Coin-tossing Problem

My friend Ben 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 sequnece are 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 percent 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 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. Norman L. Roth,
    October 17, 2016 at 2:13 pm

    Oct. 17, 2016

    Very clearly stated M. {Lars} Syll,

    Is your didactic parable epistemologically related to the famous principle stated by Ludwig V.. Mises (with obvious influence fro his brother, Richard V. Mises} : “Case probability cannot be class {not in the marxoid sense} probability” ?

    GOOGLE: {1} Norman L. Roth {2} Norman L. Roth, Technological Time {3} Norman L. Roth, Economics of work {4} Norman L. Roth. economist {5} Norman L. Roth, Origins of Markets

  2. Walter Daum
    October 17, 2016 at 2:34 pm

    On the coin-tossing problem. When confronted with a sequence like H H H H H H H H H H,
    the average person-in-the-street might say that the next toss is likely to be T, because H and T tend to balance out in the end. A mathematician would say, as you do, that if the coin is fair any sequence is as likely as any other, so H and T have equal probability on the next toss. However, a statistician might reason that the all-H sequence is unlikely with a fair coin and so would entertain the hypothesis that the coin is not fair and is biased toward H, and so would predict H on the next toss. That is, the all-H sequence is grounds for suspicion unless the fairness of the coin is somehow guaranteed.

    • Paul Schächterle
      October 17, 2016 at 9:20 pm

      Let me preface that I don’t consider these probability-related things economic issues. But they are interesting nonetheless.

      I would argue that the “H H H H …” sequence is just a very memorable sequence. Any other particular sequence could be just as “suspicious”, for example if one person always predicts correctly or another person always guesses wrong.

      • Paul Schächterle
        October 17, 2016 at 9:24 pm

        On second thought I find the argument of the statistician that the coin is (physically or technically) biased very plausible.

  3. October 17, 2016 at 3:28 pm

    I agree that people often conflate ‘ought to make you suspicious’ and probability in this way. But why?

    In the coin case I might reasonably think of a coin as having a probability of Heads having some probability distribution, with double Heads being an extreme case. As the results from the first sequence came in then, according to Bayes’ theorem’, my subjective probability ought to be increasingly skewed towards the extreme. For the second sequence they would sharpen up near 0.5. If my initial probability distribution was concentrated around 0.5 then the first sequence would seem highly suspicious, whereas the second would seem very ordinary. From this viewpoint suspicion is something to do with probability, but a different one from that in the quote above.

    From a frequentist point of view, if ‘the coin is fair-ish’ is a null hypothesis then the first sequence will be significant at the 5% level, the second not at all. So it seems to me that from either mainstream view, the first sequence is reasonably termed ‘ suspicious’, the second not. One could also look in terms of information theory, where the first sequence would be highly informative, the second not. But I agree with the headline!

    • October 17, 2016 at 5:27 pm

      I see I have inadvertently duplicated Walter’s comment. Somehow I hadn’t seen it. But to extend the point, I have previously commented that in psychology experiments psychologists assume that it is somehow rational to take psychology experimenters ‘at face value’, as if what they said was somehow ‘guaranteed’. I note that many subjects whose behaviour psychologists hold to be irrational could be explained away by supposing that they didn’t absolutely trust the experimenters. That is, they entertained some uncertainty.

      In the above sense, I am proud to be irrational.

  4. antireifier
    October 17, 2016 at 3:56 pm

    I have always thought the coin toss example was dumb. Some people hone their physical skills to the point that more often than not they exert the exact same force so if the coin is always in the same starting position, it will more often than not come up the same way. Fatigue factors set in but who is to say that they will result in different results? Now across huge groups of people maybe the actual results will conform to probability theory and approach an even distribution.

  5. October 17, 2016 at 5:59 pm

    https://arxiv.org/abs/quant-ph/0703222 there are many other discussions /solutions of ‘conjunction fallacy’/linda problem . i view myself as a ‘frequentist’ and bayes’ ‘theorem’ to be included in that—its just a rearrangement of the definition of conditional probability. (and it has an assumption it it which is almost never stated–that the unconditional joint probabilities p(a,b)=p(b,a) . there is no reason to assume that in a temporal system e.g. p(dead, born, t)=p(born, dead, t). For frequentists the baysian prior is assumed at the beginning.

    human reasoning doesn’t operate strictly using axiomatic probability calculus (for linda’s case, venn diagrams). so its equally likely lindA is a bank teller active in the feminist but looking for another job. i once worked for a major investment fund for a short period doing absolutely trivial investment startegy stuff —i just stayed long enough to collect a few paychecks.

    • October 18, 2016 at 11:17 am

      I find some of the language in the psychology experiments very loose, making it hard to interpret the findings.

  6. Rhonda Kovac
    October 18, 2016 at 3:12 pm

    The description, “Linda is a bank teller” means to most people that Linda is an ‘ordinary’ bank teller. As such, Linda is less likely to be an active feminist. And it is perfectly ‘rational’ and sensible to conclude, based on that understanding (and on the other information provided, and on knowledge of the general population), that the second description is more likely to be true than the first.

    This is a communication issue as much as it is a ‘logical’ one.

    Where was the mistake made? Was the mistake on the part of the respondent not treating the material ‘rationally’? Or was it on the part of the investigator failing to communicate the question in a manner commensurate with the mentality and understanding of the respondent?

    If a more extended effort were made by the presenter to clarify that the first description should not be treated in an ordinary sense, but rather in a strict, technical one, from which no other information about Linda should be inferred then there would be far fewer ‘irrational’ responses.

    In the real world much ‘irrational’ behavior in society comes from a failure to fully operate in alignment with people’s rational capabilities, rather than from their supposed ‘defects’ in reasoning.

  7. charlie
    October 18, 2016 at 3:32 pm

    among social scientist writings K & T’s papers have had a serious impact on my ideas re: economics I have read many of their papers. Too bad economists seem so insulated against biology, ecology and other social scientists.

  8. October 18, 2016 at 4:54 pm

    I was reminded of the extremely interesting introduction to :book entitled “Bayesian Rationality” by Oaksford and Chater. Research in Artificial Intelligence ran into a dead end, because standard logical deductive models failed to match actual human reasoning. This led to a contradiction — one the hand, it appeared that human beings are extremely irrational, on the other hand, humans were much better than computers and the logical/deductive machinery, at many kinds of unstructured tasks. It was only when the logical deductive model of reasoning was abandoned, that progress was finally made on teaching computers how to think like humans.
    So when we run into a conflict between “logic” and “human reasoning”, the fault may lie with logic. Just as the utility maximization of homo economicus labeled as “rational behavior” is actually highly irrational. — this is to amplify and support Rhonda Kovac’s views.

  9. October 18, 2016 at 7:30 pm

    In the H H H H … argument Lars’ answer is wrong. The logic is that of parallel events, not that of independent ones. To combine the probabilities one needs to invert the probabilities and then invert the sum, so 2 + 2 = 4 gives the probability of two heads appearing in succession as 1/4. Have we forgotten how often this is an issue in the question of whether or not a child will inherit a defective gene?

    • October 18, 2016 at 7:45 pm

      Since the probability of T is also 50% that may seem dubious, but the probability of 10 H’s in ten throws works out a lot less than 5 H’s out of ten throws.

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