Home > Uncategorized > Probability and rationality — trickier than most people think

Probability and rationality — trickier than most people 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 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. Ken Zimmerman
    September 4, 2021 at 1:10 am

    Let’s bring these questions down to eye level and get off our cloud.  Heuristics are the strategies derived from previous experiences with similar problems. These strategies depend on using readily accessible, and applicable, information to take on problem solving in human beings, machines and conceptual issues. When an individual applies a heuristic in practice, it generally performs as expected. However, it can at times create systematic errors or biases.  More simply, heuristics are smart  rules  of  thumb  that humans  apply  in  everyday reasoning  when  optimization,  such  as  maximizing  expected  utility,  is impossible  or  would  cost  too  much  time. Just to be clear, optimizing is never really possible. Heuristics is a big part of scientific work. Though often tacitly or submerged beneath layers of scientific procedures.

    The most fundamental heuristic is trial and error, which can be used in everything from matching nuts and bolts to finding the values of variables in algebra problems. In mathematics, some common heuristics involve the use of visual representations, additional assumptions, forward/backward reasoning, and simplification.  Here are a few commonly used heuristics from George Pólya’s 1945 book, How to Solve It.

    If you are having difficulty understanding a problem, try drawing a picture. If you can’t find a solution, try assuming that you have a solution and seeing what you can derive from that (“working backward”). If the problem is conceptual, try examining a concrete example. Try solving a more general problem first (the “inventor’s paradox”: the more ambitious plan may have more chances of success).*

    In the processes of creating culture and society, heuristics are simple, efficient rules, learned or inculcated by evolutionary processes, that have been proposed to explain how people make decisions, come to judgements, and solve problems typically when facing complex problems or incomplete information. Researchers test if people use those rules with various methods. These rules work well under most circumstances, but in certain cases can lead to systematic errors or cognitive biases.

    Psychologists tend to focus on the heuristics people use to make  judgments  or  estimates  of  probabilities  and  frequencies  in  situations  of uncertainty, which is the usual situation for humans. Most frequently used here are  the  availability,  representativeness,  and  anchoring  and adjustment  heuristics. Psychologists invented the labels. Which is one of several factors preventing psychologists from gaining a fuller understanding of how people invent and use heuristics.


    *The inventor’s paradox is a phenomenon that occurs in seeking a solution to a given problem. Instead of solving a specific type of problem, which would seem intuitively easier, it can be easier to solve a more general problem, which covers the specifics of the sought-after solution. The inventor’s paradox has been used to describe phenomena in mathematics, programming, and logic, as well as other areas that involve critical thinking.

  2. September 4, 2021 at 12:34 pm

    The inventor’s paradox. Yes, this the reason for valuing fundamental science, which resolves not just the paradigmatic problem but lots of others too.

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