Extensional Cues

EXTENSIONAL CUES

People sometimes estimate a conjunction as more likely than a category that contains it. They can nevertheless understand and apply the conjunction rule when a question makes that relationship clear. The following studies used versions of a health-survey problem to test which cues help. Participants were statistically naive students at UBC. Mean estimates are given in parentheses.

A health survey was conducted in a sample of adult males in British Columbia, of all ages and occupations. Please give your best estimate of the following values:

What percentage of the men surveyed have had one or more heart attacks? (18%)

What percentage of the men surveyed are both over 55 years old and have had one or more heart attacks? (30%)

This version of the health-survey problem produced a substantial number of conjunction errors among statistically naive respondents: 65% of the respondents (N = 147) assigned a strictly higher estimate to the second question than to the first [the incidence of the conjunction fallacy was considerably lower (28%) for a group of advanced undergraduates at Stanford University (N = 62) who had completed one or more courses in statistics]. Reversing the order of the constituents did not significantly affect the results.

The observed violations of the conjunction rule in estimates of relative frequency are attributed to the A → B paradigm. We propose that the probability of the conjunction is biased toward the natural assessment of the strength of the causal or statistical link between age and heart attacks. Although the statement of the question appears unambiguous, we considered the hypothesis that the respondents who committed the fallacy had actually interpreted the second question as a request to assess a conditional probability. A new group of UBC undergraduates received the same problem, with the second question amended as follows:

Among the men surveyed who are over 55 years old, what percentage have had one or more heart attacks?

The mean estimate was 59% (N = 55). This value is significantly higher than the mean of the estimates of the conjunction (45%) given by those subjects who had committed the fallacy in the original problem. Subjects who violate the conjunction rule therefore do not simply substitute the conditional P(B|A) for the conjunction P(A&B).

A seemingly inconsequential change in the problem helps many respondents avoid the conjunction fallacy. A new group of subjects (N = 159) were given the original questions but were also asked to assess the “percentage of the men surveyed who are over 55 years old” prior to assessing the conjunction. This manipulation reduced the incidence of conjunction error from 65% to 31%. It appears that many subjects were appropriately cued by the requirement to assess the relative frequency of both classes before assessing the relative frequency of their intersection.

The following formulation also facilitates extensional reasoning:

A health survey was conducted in a sample of 100 adult males in British Columbia, of all ages and occupations. Please give your best estimate of the following values:

How many of the 100 participants have had one or more heart attacks?

How many of the 100 participants are both over 55 years old and have had one or more heart attacks?

The incidence of the conjunction fallacy was only 25% in this version (N = 117). Evidently, an explicit reference to the number of individual cases encourages subjects to set up a representation of the problems in which class inclusion is readily perceived and appreciated. We have replicated this effect in several other problems of the same general type. The rate of errors was further reduced to a record 11% for a group (N = 360) who also estimated the number of participants over 55 years of age prior to the estimation of the conjunctive category. The present findings agree with the results of Beyth-Marom (1981), who observed higher estimates for conjunctions in judgments of probability than in assessments of frequency.

The results of this section show that nonextensional reasoning sometimes prevails even in simple estimates of relative frequency in which the extension of the target event and the meaning of the scale are completely unambiguous. However, we found that the replacement of percentages by frequencies and the request to assess both constituent categories markedly reduced the incidence of the conjunction fallacy. It appears that extensional considerations are readily brought to mind by seemingly inconsequential cues. A contrast worthy of note exists between the effectiveness of extensional cues in the health-survey problem and the relative inefficacy of the methods used to combat the conjunction fallacy in the Linda problem (argument, betting, “whether or not”). The force of the conjunction rule is more readily appreciated when the conjunctions are defined by the intersection of concrete classes than by a combination of properties. Although classes and properties are equivalent from a logical standpoint, they give rise to different mental representations in which different relations and rules are transparent. The formal equivalence of properties to classes is apparently not programmed into the lay mind.