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Matt_Simpson comments on So You Think You're a Bayesian? The Natural Mode of Probabilistic Reasoning - Less Wrong
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Here's my take on 'Linda'. Don't know if anyone else has made the same or nearly the same point, but anyway I'll try to be brief:
Let E be the background information about Linda, and imagine two scenarios:
Now obviously P(A | E) is greater than or equal to P(B | E). However, I think it's quite reasonable for P(A | E + "someone told us A") to be less than P(B | E + "someone told us B"), because if someone merely tells us A, we don't have any particularly good reason to believe them, but if someone tells us B then it seems likely that they know this particular Linda, that they're thinking of the right person, and that they know she's a bank teller.
However, the 'frequentist version' of the Linda experiment cannot possibly be (mis?)-interpreted in this way, because we're fixing the statements A and B and considering a whole bunch of people who are obviously unrelated to the processes by which the statements were formed.
(Perhaps there's an analogous point to be made about your second example: Someone being tested at all is likely to be someone for whom there are independent reasons why they might have the disease (perhaps they exhibited some of the symptoms, got worried and went to see their doctor.)
But surely the experiment must have specified that the person being tested for the disease was picked at random from the population?)
Why would you think that subjects are working from a different state of information for the two possibilities in the Linda question? Here's the question again as the subjects read it:
After reading the question, the probability of (a) and (b) is evaluated - with the same state of information: the background knowledge (E in your terms), that someone told us (a) (A), and that someone told us (b) (A). So formally, the two probabilties are:
So the conjunction rule still holds. Now it's certainly possible that subjects are interpreting the question in the way you suggest (with different states of information for A and B), but it's also possible that they're interpreting it in any number of incorrect ways. They could think it's a thinly veiled question about how they feel about feminism, for example. So why do you think the possible interpretation you raise is plausible enough to be worrisome?
note: this comment was scrapped and rewritten immediately after it was posted
Why would someone tell us "Linda is a bank teller and Linda is a bank teller and active in the feminist movement."? That would be indeed a strange sentence.
ETA: Maybe the parent comment can be formulated more clearly in the following way (using frequentist language): People parse the discussed question not as what fraction of people from category E belong also into category A?, but rather what fraction of people telling us that a person (who certainly belongs to E) belongs also to A speak truth?, or even better, what fraction of individual statements of the described type is true?
Although A may be proper subset of B, statements telling A about any particular Linda aren't proper subset of statements telling B about her. Quite contrary, they are disjoint. (That is, people tend to count frequencies of statements of given precise formulation, i.e. don't count each occurence of B as a simultaneous occurence of A, even if B can be reanalysed as A and C. Of course, I am relying on my intuition in that and can be guilty of mind projection here.)
It is entirely possible to imagine that among real world statements about former environmental activists, the exact sentence "she is a bank teller" is less often true than the exact sentence "she is a bank teller and an active feminist". I am quite inclined to believe that more detailed information is more often true than less detailed one, since the former is more likely to be given by informed people, and this mechanism may have contributed to evolution of heuristics which produce the experimentally detected conjunction fallacy.