Tuesday, January 10, 2012

The Conjunction Paradox

In Yesterday's post you were told about Susan:
Susan is 31 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-nuke demonstrations.
You were asked to rank the probabilities of certain things about Susan. Two of the choices were

a bank teller

a bank teller and an active feminist

LOGICALLY bank teller and feminist would have a HIGHER probability than bank teller and and active feminist. Some people (including me when I first was given this exercise) ranked bank teller lower bank teller and active feminist. Why? I think that either people are not good at logic in real-world situations or people implicitly view bank teller as bank teller and NOT an active feminist. This problem has been extensively studied and there are other opinions.

Of the 30 responses I got before posting this roughly 10 ranked bank teller higher than bank teller and and active feminist (which is correct), 10 ranked bank teller and and active feminist higher than bank teller, and 10 of the answers were not relevant (e.g., clarifications of the question). (One person I blocked since he explained the above and would have given away the game, and one person who left a comment 5 minutes ago I will let through but only after I post this.)

I recommend giving this exercise to students in a class that covers logic and/or probability to see what they say.

Clyde Kruskal told me about this problem. He found it here though its also at other sites. This source credits the following (which I would guess is correct). Tversky, Amos; & Kahneman, Daniel (1983), Extensional Versus Intuitive Reasoning: The Conjunction Fallacy in Probability Judgment", Psychological Review 90(4) (October): 293-315. They are famous for these sorts of psychology questions. The latter won the Nobel prize in economics for joint work with the former.

The notion that A is less likely than A AND B is called The Conjunction Fallacy. The article pointed to only gives the two choices: (1) Bank Teller, and (2) Bank Teller and an active feminist. I think its better to give all of those choices as is done here other presentations of this exercise.


  1. I refrained from replying yesterday because I was already familiar with the problem. It is a seminal result from the field of heuristics and biases (and yes, the Kahneman reference is correct).

    For light reading, I strongly recommend Daniel Kahneman's "Thinking, Fast and Slow" published last year.

    For something more scholarly, go for "Judgment under Uncertainty: Heuristics and Biases."

  2. I like this exercise, but I would use it as an opportunity to present a positive example and one to which some students in your class can relate, and by which some may feel inspired.

    For example, replace philosophy with computer science and bank teller with Google employee. Or philosophy with mathematics and bank teller with NSA employee.

    I would also avoid including pink-collar jobs/stereotypes such as "a kindergarden teacher" and "works in a bookstore and takes yoga classes", particularly in an example that uses a woman in a class that is likely male-dominated, if not in numbers, then in attitude.

  3. wasn't she 28 just a day ago?

  4. For me, I just didn't have a problem with Susan having two jobs at the same time. Bank Teller by day, active feminist in her hours off. What I thought was less likely was that she'd drift away from her passion and just be a Bank Teller by itself. But then I did presume that the descriptions were how she perceived herself and not just pure logical statements so I was one of those that essentially saw 'and NOT active feminist'. I guess it depends on how one interprets the missing context around the statements ...


  5. As the person who intuited GASARCH's intent, please let recommend the logician Raymond Smullyan's book Chess Mysteries of Sherlock Holmes, a book that (for me) charmingly illuminates both GASARCH's ranking puzzle and my own experience teaching medical diagnostic skills.

    In solving mysteries both in life and at the chessboard, Smullyan's Holmes succeeds by conceiving unifying narratives, and (for Holmes) it is a key Bayesian prior that such a narrative must always exist.

    Holmes' Bayesian presumption of a narrative prior is vital in medical diagnosis too (and IBM/Watson's computational descendents surely will incorporate Holmes' presumption). It is manifested for example in the following two clinical pearls (as they are called): (1) “When you hear hoofbeats, think 'horses' not 'zebras'”, and (2) “Never take away a patient's illusions until you can offer them better illusions.”

    At the abstract level — and seeking mathematically natural foundations for emerging discipline of narrative engineering — the cognitive processes associated to real-life narrative distillation seem (to me) naturally allied to the formal procedures that logicians call witness extraction. I have often wished that the complexity-theory blogosphere would provide (what else?) a unifying narrative about these topics.

    The process of witness/narrative extraction seems to belong to a very difficult complexity class, and yet this extraction is something we humans do always with joy, and often with facility.

    How does this work, exactly? :)

  6. I wonder how many of the incorrect answers would have been correct if you had asked simply: Is A more likely or (A and B)?

  7. I did not answer because it was obvious that the bank teller and the activist choices were the ones of interest in this trick question and this made it a bad poll. If someone is going to be an active feminist, then it IS more likely they would be a bank teller AND an active feminist than just a bank teller. Since there is no income in just being an active feminist then is IS more likely that they would be a bank teller AND an active feminist than just an active feminist since I doubt an active feminist would let her hubby support her hobby.

  8. If you carefully read the second sentence'(As a student she,,,etc)' this to me predisposes her
    activities as a student and not necessarily as a mature 31 year old adult.

    To me how one draws a conclusion is based upon how information can be presented in a biased fashion.

    For example, from the information presented is it logical to conclude that A. Susan is a single Parent, Or B. Susan is happily married with children?

  9. I think that our brain interprets the question differently than what you are claiming.

    We understand it as not "what is the probability of A assuming X?" but as "What is the probability of X assuming A?". Then putting A and B over A would make sense.

  10. I think that the reason for this paradox is that, seeing the description of Susan, the reader wants a chance to express the belief that she is a feminist. Only one of the multiple choices allows the reader to do so.

    The puzzle has nothing to do with beliefs. The reader just wants a wants to express their opinion, and one of the options is shutting them down.

  11. We are trained to solve exercise problems where most (if not all) of the data provided in the question is relevant to obtain the correct answer. That's why I believe that we intuitively think that an answer that has to do with the information in the question has higher probability than one that doesn't. Using the input question, it's more likely to believe that the right answer (to what Susan really is) must include something highly correlated to the question asked.

    For example, if we had to choose between Susan being Christian and an active feminist, I believe many would say an active feminist is more likely. Even though it's probably the other way around (at least if Susan is western as the name suggests). That's because the story is favorable to feminist and may be unfavorable to Christian.

  12. I second Pyramid Head's recommendation above. Thinking, Fast and Slow by Kahnemann was one of the best books I've read recently. It contains this example and discusses a variety of biases and fallacies that even highly trained statisticians and scientists rarely avoid.

  13. In your view an active feminist is not good at logic in a real world situation? Why?

    1. I asked this puzzle to ALL readers of my blog.
      The puzzle was ABOUT an active feminist.
      If people (like myself) get it wrong then I wonder if
      the people who got it wrong (who could be male or female)
      are bad a logic in real world situations or if they
      implicitly interpreted `bank teller' as `bank teller who is not a feminist'
      (The problem of people getting this problem wrong has been
      well studied so I am sure there are other possible explanations)

      The point is that I am reasoning about the people who got
      this puzzle wrong. I am not reasoning about or making any statement whatsoever about the fictional feminist in my problem.