Probability and rationality — trickier than most people think

26 Aug, 2021 at 18:42 | Posted in Statistics & Econometrics | 5 Comments

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 report makes you suspicious?

Most people yours truly asks this question says the first report looks suspicious.

But actually both reports 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 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. Equally probable on what measure? If the coin is fair, yes equally probable sequences. If the coin is heavily biased toward heads, then not. A likelihood ratio test on the first sequence would favor a bias for heads. But the moral is correct.

  2. Maybe behind the Linda problem lies a difference between strict logic and everyday thinking. In everyday thinking, “or” is often taken to mean “exclusive or”, as when a restaurant serves lunch with salad *or* juice, usually not both.
    Formal probability theory uses inclusive or, so that teller+feminist is a subset of teller. People untrained in probability might not see it that way, hence their “paradoxical” take on the question.

    • Interesting and confusing.
      It almost as if you can make “or” mean “and” and “and” mean “or”.

  3. The coin tossing problem creates interesting paradoxes. There are some fascinating aspects to be considered. I hope to have something written on this next week.

  4. The Linda problem illustrates a simple fallacy whose root is more basic than probability. Simple logic tells us all bank tellers who are feminists are bank tellers. No mechanism can change this fact. A Venn diagram can illustrate the simple logic underlying the probability logic quite well, revealing that the mistake is of a universal form.

    In sharp contrast, the responses to the first problem are not necessarily wrong because there are some hidden assumptions or ambiguities which are so deeply embedded that we never question them: The problem statement starts with “My friend Ben says…” What if Ben is a chronic liar so his reports follow a pattern that has nothing to do with independent coin-toss sequences and instead favor reporting the more simple patterns? this human tendency is BTW why many instances of data fraud get detected…

    And then, nowhere is it stated that these coin tosses were independent and “fair” (50% chance heads). What if Ben’s report is truthful but his tossing set-up generates H with over 50% probability? Or generates a highly autocorrelated sequence? Both make H H H H H H H H H H more probable than H T T H H T T H T H. If mechanisms such as fraudulent reporting and deviations from standard symmetry and independence assumptions are not rendered implausible by the context, there can be good reason to have all heads arouse suspicion of the presence of such mechanisms.

    Gambler’s fallacies are fallacies precisely when the mechanism used by the house (or Ben) to generate reported outcomes enforces assumptions that render the gambler’s prediction attempts futile. Those assumptions need to be stated up front to ensure we know when the gambler’s belief is fallacious, for the gambler’s betting schemes need not represent universal mistakes or irrationality, as demonstrated by exploitations of regularities in mechanical roulette wheels…

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