Why is the application of probability in quantum mechanics (QM) fundamentally different from its application in other areas? QM applies probability according to the same probability axioms as in other areas of physics, engineering, etc.
Why is there a difference?
Naively one would assume one of these possibilities:
It is not the same probability (theory?)
It is a matter of interpretation (of the formalism?)
Something else?
Many answers (which I am still studying) focus on the fact that the combined probability of two mutually-exclusive events in QM is not equal to the sum of the probabilities of each event (which holds classically by definition). This fact (appears to) makes the formulation of another probability (a quantum one) a necessity.
Yet this again breaks down to assumed independent, if this is not so, the "classic probability" is applicable (as indeed in other areas). This is one of the main points of the question.
Answer
Having given it some more thought, there is an unambiguous philosophical difference, with practical implications. The two-slit experiment provides a good example of this.
In a classical universe, any particular photon that hits the screen either went through slit A or through slit B. Even if we didn't bother to measure this, one or the other still happened, and we can meaningfully define $P(A)$ and $P(B)$.
In a quantum universe, if we didn't bother to measure which slit a photon went through, then it isn't true that it went through one slit or the other. You might say it went through both, though even that isn't entirely true; all we can really say is that it "went though the slits".
(Asking which slit a photon went through in the two-slit experiment is like asking what the photon's religion is. It simply isn't a meaningful question.)
That means that $P(A)$ and $P(B)$ just don't exist. Here's where one of the practical implications comes in: if you don't understand QM properly [I'm lying a bit here; I'll come back to it] then you can still calculate a probability that the particle went through slit A and a probability that it went through slit B. And then when you try to apply the usual mathematics to those probabilities, it doesn't work, and then you start saying that quantum probability doesn't follow the same rules as classical probability.
(Actually what you're really doing is calculating what the probabilities for those events would have been if you had chosen to measure them. Since you didn't, they're meaningless, and the mathematics doesn't apply.)
So: the philosophical difference is that when studying quantum systems, unlike classical systems, the probability that something would have happened if you had measured it is not in general meaningful unless you actually did; the practical implication is that you have to keep track of what you did or did not measure in order to avoid doing an invalid calculation.
(In classical systems most syntactically valid questions are meaningful; it took me some time to come up with the counter-example given above. In quantum mechanics most questions are not meaningful and you have to know what you're doing to find the ones that are.)
Note that keeping track of whether you've measured something or not is not an abstract exercise restricted to cases where you are trying to apply probability theory. It has a direct and concrete impact on the experiment: in the case of the two-slit experiment, if you measure which slit each photon went through, the interference pattern disappears.
(Trickier still: if you measure which slit each photon went through, and then properly erase the results of that measurement before looking at the film, the interference pattern comes back again.)
PS: it may be unfair to say that calculating a "would-have" probability means that you don't understand QM properly. It may simply mean that you're consciously choosing to use a different interpretation of it, and prefer to modify or generalize your conception of probability as necessary. V. Moretti's answer goes into some detail about how you might go about doing this. However, while this sort of thing is interesting, it does not appear to me to be of any obvious use. (It isn't clear that it gives any insight into the disappearance and reappearance of the interference pattern as described above, for example.)
Addendum: that has become clearer following the discussion in the comments. It seems that it is thought that the alternative formulation may have advantages when dealing with more complicated scenarios (QFT on curved spacetime was mentioned as one example). That is entirely plausible, and I certainly don't mean to imply that the work lacks value; however, it is still not clear to me that it is pedagogically useful as an alternative to the conventional approach when learning basic QM.
PPS: depending on interpretation, there may be other philosophical differences related to the nature or origin of randomness. Bayesian statistics is broad enough, I believe, that these differences are not of any great importance, and even from a frequentist viewpoint I don't think they have any practical implications.
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