I got to shadow some breast physicians last month, and although it's sort of off topic I think I gained some insight as to why so many doctors get this question wrong.
Which is because it's very different from any situation they ever come across in clinical practice. Guidelines are to screen people with mammography and examination; anyone who comes up as suspicious on those two tests then gets a biopsy. No one gets diagnosed with breast cancer from a mammogram alone, the progression from mammogram on to the next step is hard-coded into a pre-determined algorithm, and so the question of "This woman got a positive on the mammogram; does she have cancer?" never comes up. A question that does come up a lot is a woman panicking because she got a positive mammogram and demanding to know whether she has breast cancer, and the inevitable answer is "We'll need to do more tests, but don't worry too much yet because most of these things are false positives."
So the doctors involved know that most real mammogram results are false positives, they know how to diagnose breast cancer based on the combination of tests they actually do, they just can't do Bayesian math problems when given probabilities. This is kind of interesting if you're curious about their intelligence but as far as I know doesn't really affect clinical care.
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I agree with previous comments about publishing in journals being an important status issue, but I think there is other value as well which is being ignored. For all of its annoyances and flaws, one good thing about peer review is that it really makes your paper better. When you submit a pretty good paper to a journal and get back the "revise and resubmit" along with the detailed list of criticisms and suggestions, then by the time the paper actually makes it into the journal, chances are that it will have become a really good paper.
But to return to the issue of papers being taken more seriously when published in a journal, I think that this view is actually quite justified. For researchers who are not already very knowledgeable in the precise area that is the topic for a given paper, whether or not the paper has withstood peer review is a very useful heuristic cue toward how much weight you should place on it. Basically, peer review keeps the author honest. An author posting a paper on his website can say pretty much whatever he wants. One of the purposes of peer reviewers is to make sure that the author isn't making unreasonable claims, mischaracterizing theoretical positions, "reviewing" the relevant previous literature in a grossly selective way, etc. Like I said, if someone is already very familiar with the area, then they can evaluate these aspects of the paper for themselves. But if you'd like to communicate your position to a wider academic audience, peer review can help carry your paper a longer way.