The base rate fallacy has been considered to result from people's tendency to ignore the base rates given in tasks. In the present article, we note a particular, common structure of the tasks (the imbalanced probability structure) in which the fallacy is often observed. The equiprobability hypothesis explains the mechanism that produces the fallacy. This hypothesis predicts that task material that overrides people's default equiprobability assumption can facilitate normative Bayesian inferences. The results of our two experiments strongly supported this prediction, and none of the alternative theories considered could explain the results.
(ProQuest: ... denotes formulae omitted.)
Since a seminal study by Kahneman and Tversky (1973), a vast number of studies have been concerned with the controversial topic of base rate neglect (for reviews, see, e.g., Barbey & Sloman, 2007a; Koehler, 1996). People are supposed to be insensitive to base rate information when they engage in a task such as the following, modified from Eddy (1982, pp. 251-254; underlines and brackets will be explained later):
Recently, an incurable disease called X syndrome has begun to be reported. X syndrome is known to show symptoms similar to a cold. Suppose you are a doctor working in a hospital. You are expected to make a judgment about whether or not a patient is infected by X syndrome based on the following information.
The prevalence of X syndrome is 1%. A patient who is infected with X syndrome has an 80% chance of testing positive [having a cough]. However a patient who is not infected with the syndrome has a 9.6% chance of testing positive [having a cough]. Now a patient tests positive [has a cough]. What is the chance that the patient is actually infected with X syndrome? Please answer intuitively. _____%
(Correct Answer: 7.8%)
This is called a base rate task (or a BR task, hereafter). Let P(H) be the base rate (prevalence) of having the disease (i.e., the hypothesis) and P(D) be the probability that a person tests positive (i.e., data). This is a Bayesian inference task to derive the posterior probability of X syndrome, P(H|D), from the true positive (detectability) rate, P(D|H). There is a relationship between the true positive rate and the posterior probability, as follows:
Here, P(D) is expressed as
Therefore, the correct answer for the task is
From Equation 1, the low base rate, P(H), lowers the posterior probability, P(H|D). However, modal responses in typical experiments are about 80% (see, e.g., Bar-Hillel, 1980), the same rate as the true positive rate. This pervasive and robust phenomenon has been called base rate neglect (or base rate fallacy), because it was believed to demonstrate people's insensitivity to the base rate information.
In considering whether and why the base rate is neglected, we should ask what base rate "neglect" means. As long as a person makes a response, some value must be allocated-at least implicitly-to the base rate, and it is not in this sense ignored. As is obvious from Equation 1, if we suppose P(H|D) = P(D|H), which is a common answer, we can derive P(H) = P(D). This is exactly what base rate neglect implies. If participants assume that the probability of "testing positive" and the probability of "having X syndrome" are equal, they will give the true positive rate as an answer for a posterior probability. In our view, base rates are not ignored, but people make a default equiprobability assumption, P(H) = P(D). Thus, we propose that participants do not neglect the base rate, but that they infer that the posterior and the true positive rates are almost equal by postulating near equality in the marginal probabilities of the two target events that are focused on in the task.
BR tasks share some characteristics that we regard as essential in leading people to the fallacy. The most important feature, we believe, is the fact that the marginal probabilities, P(H) and P(D), differ greatly. …