Recent data in studies by McCloy and colleagues (McCloy et al. Reference McCloy, Beaman, Morgan and Speed2007; McCloy et al., submitted) show that judgment of cumulative, disjunctive risk (i.e., the probability of avoiding an adverse event over a period of time during which one continually engages in a risky activity) benefits from presentation in a frequency, rather than a probability, format (McCloy et al., submitted). It does this in a similar manner to the way in which judgments of conditional probability avoid the base-rate neglect fallacy if presented in natural frequency format (Gigerenzer & Hoffrage Reference Gigerenzer and Hoffrage1995). Further, training in translation from probability to frequency formats shows similar improvements relative to baseline for both types of judgment (McCloy et al. Reference McCloy, Beaman, Morgan and Speed2007). However, the effects of both format and training are mimicked by presenting information in a partitive or “nested” set structure (in our studies, diagrammatically represented by probability trees rather than Euler circles). This suggests that similar processes may be involved in both problem types, and we applaud Barbey & Sloman (B&S) for attempting to break down the nature of those processes rather than remaining satisfied with a “natural frequency” label. However, we do not believe that B&S have (yet) produced a full and complete account of the means by which dual processes may produce rationality within certain given ecologies.
One worry is the assertion that people do not have an (evolved or otherwise) capacity to encode frequency information. This is a claim concerning a failure to observe a particular capability and might reflect failures in the observational technique employed as much as a failure in the capability itself. The data reviewed by B&S are based upon individuals' inability to produce, on demand, explicit knowledge obtained of frequencies from episodes of incidental learning. This is known to be problematic (Morton Reference Morton1968). This may – perhaps – explain why well-controlled laboratory studies (e.g., Sedlmeier et al. Reference Sedlmeier, Hertwig and Gigerenzer1998) have had better success in showing accurate frequency judgments than studies in naturalistic settings do, although, as B&S note, such lab studies have typically not covered autobiographical events to which a Bayesian inference structure can be applied.
A second concern is directly related to the issue of training individuals to translate from probabilities to frequencies (McCloy et al. Reference McCloy, Beaman, Morgan and Speed2007). In this study we examined whether training people to represent the data to themselves in a partitive structure allowed for accurate responding to cumulative risk judgments. This was contrasted with equivalent judgment based on single-event probabilities expressed relative to a single time period. Participants were given statements such as: “Suppose that a person who drives fast whilst using a cell phone has a 90% probability of not being involved in a car accident in any one year. What is the probability that they avoid being involved in an accident at all if they continue to drive in the same way, over the same roads, for three years?” This statement produced, on average, only 25% correct responding, although the statistical “rule” for disjunctive cumulative probability (1 – pn) is considerably simpler in form than Bayes' theorem. Following training in recoding this data into either a probability or a frequency tree, however, performance improved to approximately 67% correct, and stayed at this level after a one-week interval. This suggests that problem structure is important for more than just base-rate neglect problems – but it also begs the question of how problem structure is represented if not in frequency terms. Our participants were taught to use tree-structures and proved reasonably able at learning and using these. Sloman et al. (Reference Sloman, Over, Slovak and Stibel2003) employed Euler circles to likewise make nested set relations transparent for Bayes' theorem. Unfortunately, although tree structures and Euler circles may be interchangeable as aids to conditional reasoning, they are not necessarily equivalent when applied to cumulative probability judgments, as the diagram in Figure 1 makes clear.
Figure 1. Three situations (a, b, and c) representing the relationship between the set of car accidents in year 1 (solid circle, SC) and year 2 (dashed circle, DC).
With tree structures, the branching of different states of the world over time can be represented within a single tree regardless of whether the problem is conditional or cumulative (a fact that caused our participants some difficulty when unexpectedly confronted with problems of an unexpected type). In contrast, for conditional probabilities the number of (or chances of) having a disease, given a positive test, is represented by only one possible set of Euler circles (see target article's Fig. 1). Following Sloman et al. (Reference Sloman, Over, Slovak and Stibel2003), the probability of avoiding a car accident over the two-year period is 1 minus p (SC union DC) but the number of (or chances of) avoiding a car crash over a two-year period is, potentially, the complement of any one of three pairs of Euler circles (a, b, and c in our Fig. 1). Although for any realistic probability values circle b is considerably more probable, this may not be immediately apparent using solely partitive information.
Despite this difficulty, framing cumulative probability judgments in such a way that nested relations are transparent improves performance in a manner similar to making sorts (or kinds) of relations transparent within Bayesian judgments. This leaves us with the questions: If not all diagrammatic representations of nested relations are equal, what type of mental representation(s) of such relations are being employed in order to reason extensionally; and, crucially, when and how is this representational system employed?
