Conjunction Junction

Robert thinks behavioral psychology experiments, and the papers they spawn, can tend toward the gobbledygook. But they do make fun discussion around the dinner table and they often give Robert ammunition he uses to tease his wife. Last night’s teasing was based around the Linda the Bank Teller problem invented by the “famous” Tversky and Kahneman  1983 paper discussing the so-called “conjunction fallacy.”  Robert is not smart enough to not be skeptical of the papers in this category.

These papers all seem to take the same approach.  First, find some math skills problem that people are generally bad at solving (while someone in a white coat watches on). Then come up with a name for that particular shortcoming that includes the word “fallacy” or “error” so that it then be discussed as some sort of core aspect of human nature.

But the experiments are fun stuff and are enough to get Robert thinking. Plus, it’s fun to read articles by smart guys.  Which Tversky and Kahneman surely are.

The core question in the paper can, perhaps, be described as follows.

Why do most people answer the following two questions differently?


Linda is a 31-year-old woman.

Which of the following is more probable?

A. Linda is a bank teller.

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


Linda 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-nuclear demonstrations.

Which of the following is more probable?

A. Linda is a bank teller.

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

The answer to both questions is A.  But lots of people, almost irrespective of the number of years of their schooling, answer A in Question 1 but wrongly answer B in Question 2. We can say that more than 80% of people not trained in statistics do this.

Gobbledygook Alert: The answer cannot be B.  The reason? B is a “conjunction,” and therefore cannot be more probable than one of its constituents. And, in case you didn’t notice, answer A is one of its constituents.  In other words, answer B includes more conditions (or, stated yet another way, is more limiting than) answer A. B is an extension of A.

Note: There are some other very smart psychologists/economists/statisticians who believe the answer CAN be B, but I’m going to ignore them because I understand them less than the inventors of the Linda problem.

Anyway, Robert is usually no better at these damn problems that the rest of the world, but he has bragging rights with this one. Thirty years ago he spent about 4 months studying for the LSAT, which is a four-hour test asking dozens and dozens of questions like these.  Which means he can tease Mira a little more when she gets it wrong.

The fun part is talking about WHY everyone gets it wrong.  That’s the mystery. Of course, on one level, it is obvious that all the stuff about Linda’s education and college life in the preamble to Question 2 leads people to give the wrong answer. But what, exactly, causes the confusion? What are people thinking when they give the wrong answer? In the words of the authors, “Why do intelligent and reasonably well-educated people fail to recognize the applicability of the conjunction rule in transparent problems?” Note, the word “transparent” is pretty loaded.

But Robert’s favorite part of the paper is when the authors talk about the post-experimental interviews and discussions with college undergraduate (i.e., naive) and trained graduate (i.e., sophisticated) students.

The authors say that naive as well as sophisticated subjects generally noticed the nesting of conditions in the test, but the naive, unlike the sophisticated, did not appreciate its significance for probability assessment.  On the other hand, most naive subjects did not attempt to defend their responses.  As one subject said after acknowledging the validity of the conjunction rule, “I thought you only asked for my opinion.”

According to the authors:

“The interviews and the results of the direct transparent tests indicate that naive subjects do not spontaneously treat the conjunction rule as decisive.  Their attitude is reminiscent of children’s responses in Piagetian experiment.  The child in the preconservation state is not altogether blind to arguments based on conservation of volume and typically expects quantity to be conserved (Bruner, 1966). What the child fails to see is that the conservation argument is decisive and should overrule the perceptual impression that the tall container holds more water than the short one. Similarly, naive subjects generally endorse the conjunction rule in the abstract, but their application of this rule to the Linda problem is blocked by the compelling impression that Teller-plus-Feminist is more representative of her than Teller is.  In this context, the adult subjects reason as if they had not reached the stage of formal operations.  A full understanding of a principle of physics, logic, or statistics requires knowledge of the conditions under which it prevails over conflicting arguments, such as the height of the liquid in a container or the representativeness of an outcomeThe recognition of the decisive nature of rules distinguishes different developmental stages in studies of conservation; it also distinguishes different levels of statistical sophistication.”

Nice paragraph.

Here’s a fun video showing a child in a “preconservation state.” Robert believes it nicely demonstrates how children at this stage have general notions about the rules of physical conservation, but don’t bring that understanding to the forefront of their minds in a decisive way when called to do so.  They also do not resist when confronted with their violation of the rules. According to the authors, it’s the same for “naive” adults and statistics. It is not true for Mira, who is good at resisting.