Basic understanding of posterior probability

Consider the following task[TaskA]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 withtheanomaly.Ifawomanhasachildwiththeanomaly,thereisa75%chancethatshehasapositivetest result. If she does not have a child with the anomaly, there is still a 12.5% chance that she hasa 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?Toanswercorrectly,onehastointegratethepriorprobabilitythatawomanhasachildwiththeanomaly (i.e., the prevalence rate: 4%) with information about the test’s statistical properties. OnthebasisofthisinformationandtheevidencethatEmmatestedpositive,onecanproduceacorrectposterior evaluation by computing the ratio:Probability(Anomaly|PositiveTestResult)=Probability(“PositiveTestResultandAnomaly”)/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 theprevalencerateandthefalsepositiverate(i.e.,96%×12.5%=12%),andthenaddittotheinitiallyobtainedvalue(i.e.,3%+12%=15%).Veryfewrespondents,includinghealth-careprofessionals,produce the correct probability ratio (i.e., 3%/15% = 20%). Failures to solve tasks of this sortlead to pessimistic conclusions about naive probabilistic reasoning (e.g., Casscells et al., 1978).Subsequent studies, however, licensed more optimistic conclusions, showing that some versionsof these tasks led to better performances. About half of the respondents succeed when reasoningwith natural frequencies (e.g., “Three out of the 4 women who had a child with the anomaly had apositive test result”) or numbers of chances (e.g., “In 3 out of the 4 chances of having a child withthe anomaly the test result is positive”; see, respectively, Hoffrage and Gigerenzer, 1998; Girottoand Gonzalez, 2001). On the basis of these results, the current, common account is that posteriorprobability reasoning improves in versions that allow respondents to both rely on an appropriaterepresentation of subsets of countable elements (e.g., observations, tokens), and to easily associateposterior evidence with one of these subsets (Barbey and Sloman, 2007).A generally unnoticed aspect of the results mentioned above is that they concern educatedrespondents, like undergraduates and physicians, and that only about half of these respondentsbenefit from the simplified versions of the tasks. Even more unnoticed is the fact that respondentssampled from the general public do not benefit at all from these versions. Indeed, in samples ofpregnantwomen,manyofwhomhadahighschoollevelofeducationorless,almostallrespondentsfailedtocomputethecorrectprobabilityratio,eveniftheyhadtoreasonaboutnaturalfrequencies(Bramwell et al., 2006) or numbers of cases (Pighin et al., 2015). In other words, they failedtasks that, in principle, should have activated the appropriate set representation. Their failure isstriking because, unlike the participants of previous studies who had to reason about hypothetical