I bet you would. It wouldn't have to be an unsolved problem; one to which they couldn't too-quickly work out the answer would suffice. The sort of problem you'd need would be one for which there's a plausible-seeming argument for each of two conclusions -- e.g., the "Feynman sprinkler" problem -- and then you'd frame the question so as to suggest one or other of the arguments.
But it would be disappointing and surprising if physics professors turned out to do no better at such questions than people with no training in physics.
(If you make the question difficult enough and give them little enough time, that might happen. Maybe the Feynman sprinkler problem with 30 seconds' thinking time would do. Question: How closely analogous is this to the trolley problem for philosophers? Question: If you repeat the study we're describing here but encourage the philosophers to spend several minutes thinking about each question, do the framing effects decrease a lot? More or less than for people who aren't professional philosophers?)
Imagine someone finding out that "Physics professors fail on basic physics problems". This, of course, would never happen. To become a physicist in academia, one has to (among million other things) demonstrate proficiency on far harder problems than that.
Philosophy professors, however, are a different story. Cosmologist Sean Carroll tweeted a link to a paper from the Harvard Moral Psychology Research Lab, which found that professional moral philosophers are no less subject to the effects of framing and order of presentation on the Trolley Problem than non-philosophers. This seems as basic an error as, say, confusing energy with momentum, or mixing up units on a physics test.
Abstract:
We examined the effects of framing and order of presentation on professional philosophers’ judgments about a moral puzzle case (the “trolley problem”) and a version of the Tversky & Kahneman “Asian disease” scenario. Professional philosophers exhibited substantial framing effects and order effects, and were no less subject to such effects than was a comparison group of non-philosopher academic participants. Framing and order effects were not reduced by a forced delay during which participants were encouraged to consider “different variants of the scenario or different ways of describing the case”. Nor were framing and order effects lower among participants reporting familiarity with the trolley problem or with loss-aversion framing effects, nor among those reporting having had a stable opinion on the issues before participating the experiment, nor among those reporting expertise on the very issues in question. Thus, for these scenario types, neither framing effects nor order effects appear to be reduced even by high levels of academic expertise.
Some quotes (emphasis mine):
When scenario pairs were presented in order AB, participants responded differently than when the same scenario pairs were presented in order BA, and the philosophers showed no less of a shift than did the comparison groups, across several types of scenario.
[...] we could find no level of philosophical expertise that reduced the size of the order effects or the framing effects on judgments of specific cases. Across the board, professional philosophers (94% with PhD’s) showed about the same size order and framing effects as similarly educated non-philosophers. Nor were order effects and framing effects reduced by assignment to a condition enforcing a delay before responding and encouraging participants to reflect on “different variants of the scenario or different ways of describing the case”. Nor were order effects any smaller for the majority of philosopher participants reporting antecedent familiarity with the issues. Nor were order effects any smaller for the minority of philosopher participants reporting expertise on the very issues under investigation. Nor were order effects any smaller for the minority of philosopher participants reporting that before participating in our experiment they had stable views about the issues under investigation.
I am confused... I assumed that an expert in moral philosophy would not fall prey to the relevant biases so easily... What is going on?