So What's Going On With The CAL-101

450). This ��base-rate neglect�� is one of the prime examples for a cognitive fallacy investigated in the ��heuristics and biases�� program (Kahneman et al., 1982), and Bar-Hillel (1980) stated that ��the genuineness, the robustness, and the generality of the base-rate fallacy are matters of established fact�� (p. 215). This conclusion has been challenged by Gigerenzer and Hoffrage (1995) with a study in which they represented the information about base rate and the two likelihoods in terms of natural frequencies. Using this representation format, our task reads as follows: The Skiwell Manufacturing Company gets material from two suppliers. Out of 1,000 items, supplier A delivers 300 and supplier B delivers the remaining ones. Past records indicate that 45 of the 300 items delivered by supplier A are defective and that 70 out of the 700 items delivered by B are defective. Since it is impossible to tell which supplier the material came from once they are in the inventory, the manager wants to know: How many of the items that have been identified as defective come from supplier A? Natural frequencies are the frequencies that naturally result if a sample is taken from a population (or if the entire population is considered). In case of one hypothesis (H, with its complement �CH) and one dichotomous, diagnostic variable that represents the data (D), natural frequencies are the four entries in the bivariate 2 �� 2 table. The frequencies of the four conjunctive events can be displayed in two trees, in each of which the total sample size (or population) is the top node, and the four possible combinations are on the lowest level. One of these two possible trees displays the row margins at the intermediate level, and the other one the column margins. For instance, in Figure ?Figure1,1, Panel B, the two natural frequencies for a sample of 1,000 items are displayed at the intermediate level: 300 come from supplier A and 700 from supplier B, corresponding to the two base rates of 30 and 70%. From this tree in Panel B, it is relatively easy to determine the total number of defective items (45 + 70 = 115), and the total number of intact items (255 + 630 = 885). These two numbers are basically the margins of the diagnostic variable, and the first is included in the Bayesian solution to our task: Of the 115 defective items, 45 were delivered by supplier A. This is also the Bayesian response that we withheld above when we presented the SKAP1 problem in terms of percentages: p(H|D) = 0.39 (or, as a ratio, 45/115). From Figure ?Figure1,1, Panel B, it is also easy to construct the tree displayed in Panel C, which would also allow one to answer to other questions, for instance, how many of the intact items were delivered by supplier B. FIGURE 1 Numerical information of the Skiwell Manufacturing Company task. (A) Information provided in percentages.