Basic understanding of posterior probability

Consider the following task [Task A] A prenatal test determines whether an unborn child has a chromosomal anomaly. A priori, namely, before undergoing the test, a pregnant woman has a 4% chance of having a child with the anomaly. If a woman has a child with the anomaly, there is a 75% chance that she has a positive test result. If she does not have a child with the anomaly, there is still a 12.5% chance that she has a positive test result. Emma, a pregnant woman, undergoes a prenatal test. The result is positive. What is the probability that she has a child with the anomaly? To answer correctly, one has to integrate the prior probability that a woman has a child with the anomaly (i.e., the prevalence rate: 4%) with information about the test's statistical properties. On the basis of this information and the evidence that Emma tested positive, one can produce a correct posterior evaluation by computing the ratio: Probability (Anomaly|Positive Test Result) = Probability (" Positive Test Result and Anomaly ")/ Probability (" Positive Test Result "). To obtain the numerator, one has to combine the prevalence rate and the test's sensitivity rate (i.e., 4% × 75% = 3%). To obtain the denominator, one has to combine the complement of the prevalence rate and the false positive rate (i.e., 96% × 12.5% = 12%), and then add it to the initially obtained value (i.e., 3% + 12% = 15%). Very few respondents, including health-care professionals, produce the correct probability ratio (i.e., 3%/15% = 20%). Failures to solve tasks of this sort lead to pessimistic conclusions about naive probabilistic reasoning (e.g., Casscells et al., 1978). Subsequent studies, however, licensed more optimistic conclusions, showing that some versions of these tasks led to better performances. About half of the respondents succeed when reasoning with natural frequencies (e.g., " Three out of the 4 women who had a child with the anomaly had a positive test result ") or numbers of chances (e.g., " In 3 out of the 4 chances of having a child with the anomaly the test result is positive " ; see, respectively, Hoffrage and Gigerenzer, 1998; Girotto and Gonzalez, 2001). On the basis of these results, the current, common account is that posterior probability reasoning improves in versions that allow respondents to both rely on an appropriate representation of subsets of countable elements (e.g., observations, …