Barbey & Sloman's (B&S's) comprehensive analysis of the literature shows that, given an accurate set representation, naive individuals are able to evaluate posterior probability by performing elementary set operations on various information types (e.g., natural frequencies or numbers of chance). We agree with B&S's conclusion that natural frequency theorists have raised a deep and fundamental question: What are the conditions that compel individuals to reason extensionally? However, we argue that natural frequency theorists have not used the appropriate methodology to answer this question.

It is ironic that the hypothesis “the mind is a frequency monitoring device” has been tested almost exclusively by means of complex word problems in which numerical information is conveyed by symbols. There are three reasons to doubt the suitability of this approach. First, verbal expressions of frequencies are potentially ambiguous. For instance, in some reputedly natural frequency problems, the statements are stated in the future tense: “8 out of every 10 women with breast cancer will get a positive mammography” (Gigerenzer & Hoffrage Reference Gigerenzer and Hoffrage1995). The point is that statements of this sort express expected rather than observed frequencies, and could be correctly interpreted as statements about probabilities. For example, the above statement could be understood as follows: “Women with breast cancer have 8 chances in 10 of getting a positive mammography.” To make this point clearer, consider another statement that expresses expected frequencies: “This coin will land heads up one out of two times.” Given that it is stated in the future tense, this statement does not mean that an individual has tossed the coin twice and observed that one of the times it landed heads up, but rather, that the coin has a priori one out of two chances of landing heads up. If readers interpret predicted frequencies as probabilities, answers based on information understood as chance data could be mistaken for answers based on natural frequencies. Second, Bayesian reasoning concerns the revision of probability in the light of new evidence. However, the standard problems used in the literature typically ask for a single judgment, rather than for two successive judgments, the first one based on prior information and the following one on posterior information. Hence, these problems do not investigate the ways in which individuals actually revise their judgment in light of new information. Third, word problems asking for a numerical judgment cannot be used with individuals lacking basic verbal and numerical skills, such as young children. In sum, despite their common use, standard word problems are not the best tool to seriously test general hypotheses about the nature of human judgment.

Consider a situation in which participants are presented with a bag containing four round chips (all black) and four square chips (three white and one black). Before the experimenter draws a chip from the bag, participants have to make a prior bet on the drawing of a black versus a white chip. After this first bet, the experimenter draws a chip and, keeping it in her hand, informs participants that it is square, and asks them to make another, posterior bet. A task of this sort does not present the aforementioned three weaknesses. First, it does not have the potential ambiguities of verbal statements: it certainly asks individuals to reason about a set of prior possibilities, not about observed frequencies in a series of actual draws. Second, it requires participants to update their choice in light of new evidence. Third, it can be used to investigate whether young children, who cannot tackle complex word problems, are nonetheless able to use new evidence for evaluating an uncertain event.

Testing whether children possess some intuitions of posterior probability may sound paradoxical, given the difficulties of adults' reasoning discussed in B&S's review. But, if reasoning about uncertain events depends on the application of elementary set operations, then even children should be able to solve posterior probability tasks, at least from the age at which they are able to compare and add quantities. Indeed, Girotto and Gonzalez (in press, Study 1) have shown that from the age of about five, children perform correctly in the chip task. As found in previous studies (e.g., Brainerd Reference Brainerd1981), children first answered “black,” by reasoning about the initial set of possibilities in which there were five black and three white chips. Then they correctly updated their initial choice, by considering the subset of possibilities compatible with the new piece of information (i.e., the four squares). In sum, preschoolers are able to apply correct extensional procedures in reasoning about the random events produced by a chance device. But they do so even when they have to reason about a single, not repeatable event produced by an intentional agent (Girotto & Gonzalez, in press, Study 3). For example, children were presented with two boxes, each containing three animals (two cats and one dog vs. two dogs and one cat). The experimenter informed children that a troll secretly put one chocolate in the bag of one animal. Children had to choose a box in order to find the animal with the chocolate. There was no optimum choice, given that prior evidence did not favor one box over the other. After they made their choice, children were informed that a cat carried the chocolate and were asked to make a new choice. As predicted by the extensional reasoning hypothesis, children passed the test: even children who initially did not choose the more advantageous box, now chose the box favored by posterior evidence.

In sum, humans may be “developmentally and evolutionarily prepared to handle natural frequencies” (Gigerenzer & Hoffrage Reference Gigerenzer and Hoffrage1999, p. 430). However, they are not blind to single-case probability. Even preschoolers correctly draw posterior probability inferences about single events in nonverbal tasks asking for a choice or a non-numerical judgment. And they do so in the same situations in which adults succeed – that is, when they have to make a simple enumeration of possibilities.