Transparent Tests of the Conjunction Rule

Transparent Tests

The conjunction rule says that two events occurring together cannot be more probable than either event alone. We tried a series of increasingly desperate manipulations to induce subjects to obey it. We first presented a description of Linda that made her sound like an active feminist to a group of 142 undergraduates at UBC and asked them to check which of two alternatives was more probable:

• Linda is a bank teller. (T)
• Linda is a bank teller and is active in the feminist movement. (T&F)

The order of alternatives was inverted for half of the subjects, but this manipulation had no effect. Overall, 85% of respondents indicated that T&F was more probable than T, in a flagrant violation of the conjunction rule.

Although subjects do not spontaneously apply the conjunction rule, perhaps they can recognize its validity. We presented another group of UBC undergraduates with the description of Linda, followed by the two statements, T and T&F, and asked them to indicate which of the following two arguments they found more convincing:

Argument 1. Linda is more likely to be a bank teller than she is to be a feminist bank teller, because every feminist bank teller is a bank teller, but some women bank tellers are not feminists, and Linda could be one of them.

Argument 2. Linda is more likely to be a feminist bank teller than she is likely to be a bank teller, because she resembles an active feminist more than she resembles a bank teller.

The majority of subjects (65%, n = 58) chose the invalid resemblance argument (Argument 2) over the valid extensional argument (Argument 1). Thus, a deliberate attempt to induce a reflective attitude did not eliminate the appeal of the representativeness heuristic.

We made a further effort to clarify the inclusive nature of the event T by representing it as a disjunction. [Note that the conjunction rule can also be expressed as a disjunction rule P(A or B) ≥ P(B).] The description of Linda was used again, with a 9-point rating scale for judgments of probability, but the statement T was replaced by

Linda is a bank teller whether or not she is active in the feminist movement. (T∗)

This formulation emphasizes the inclusion of T&F in T. Despite the transparent relation between the statements, the mean ratings of likelihood were 5.1 for T&F and 3.8 for T∗ (p *< .*01, by t test). Furthermore, 57% of the subjects (n = 75) committed the conjunction fallacy by rating T&F higher than T∗, and only 16% gave a lower rating to T&F than to T∗.

The violations of the conjunction rule in direct comparisons of T&F to T∗ are remarkable because the extension of “Linda is a bank teller whether or not she is active in the feminist movement” clearly includes the extension of “Linda is a bank teller and is active in the feminist movement.” Many subjects evidently failed to draw extensional inferences from the phrase “whether or not,” which may have been taken to indicate a weak disposition. This interpretation was supported by a between-subjects comparison, in which different subjects evaluated T, T∗, and T&F on a 9-point scale after evaluating the common filler statement, “Linda is a psychiatric social worker.” The average ratings were 3.3 for T, 3.9 for T∗, and 4.5 for T&F, with each mean significantly different from both others. The statements T and T∗ are, of course, extensionally equivalent, but they are assigned different probabilities. Because feminism fits Linda, the mere mention of this attribute makes T∗ more likely than T, and a definite commitment to it makes the probability of T&F even higher!

Modest success in loosening the grip of the conjunction fallacy was achieved by asking subjects to choose whether to bet on T or on T&F . The subjects were given Linda’s description, with the following instruction:

If you could win $10 by betting on an event, which of the following would you choose to bet on? (Check one).

The percentage of violations of the conjunction rule in this task was “only” 56% (n = 60), much too high for comfort but substantially lower than the typical value for comparisons of the two events in terms of probability. We conjecture that the betting context draws attention to the conditions in which one bet pays off whereas the other does not, allowing some subjects to discover that a bet on T dominates a bet on T&F .

The respondents in the studies described in this section were statistically naive undergraduates at UBC. Does statistical education eradicate the fallacy? To answer this question, 64 graduate students of social sciences at the University of California, Berkeley, and at Stanford University, all with credit for several statistics courses, were given the rating-scale version of the direct test of the conjunction rule for the Linda problem. For the first time in this series of studies, the mean rating for T&F (3.5) was lower than the rating assigned to T (3.8), and only 36% of respondents committed the fallacy. In this statistically trained group, a majority conformed to the conjunction rule in a transparent test, although violations remained fairly common.

Elsewhere (Kahneman & Tversky, 1982a), we distinguished between positive and negative accounts of judgments and preferences that violate normative rules. A positive account focuses on the factors that produce a particular response; a negative account seeks to explain why the correct response was not made. The positive analysis of the Bill and Linda problems invokes the representativeness heuristic. The stubborn persistence of the conjunction fallacy in highly transparent problems, however, lends special interest to the characteristic question of a negative analysis: Why do intelligent and reasonably well-educated people fail to recognize the applicability of the conjunction rule in transparent problems? Postexperimental interviews and class discussions with many subjects shed some light on this question. Naive as well as sophisticated subjects generally noticed the nesting of the target events in the direct–transparent test, but the naive, unlike the sophisticated, did not appreciate its significance for probability assessment. However, 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.”

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 a Piagetian experiment. The child in the preconservation stage 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 T&F is more representative of her than T 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 outcome. The recognition of the decisive nature of rules distinguishes different developmental stages in studies of conservation; it also distinguishes different levels of statistical sophistication in the present series of studies.