Rather, a closer situation would be to monitor, say, several species' morphological mutation rate in response to specific conditions, and attempting to infer a global "probability" (in time terms as well) that such a "successful" adaptation can occur for a generic species. This will prove difficult, but is what is required for the question at hand.
Such experiments as you describe do occur, and are occurring right now. If for instance, a bacteria that evolves a certain adaptation can be compared with a frozen ancestor to see how often those ancestors subsequently evolve that same adaptation. There comes a point where evolving a certain adaptation becomes almost inevitable.
Given what we know about how the genotype becomes the phenotype, we know that it is essentially impossible to derive any general rule about any specific adaption since what happens in the future evolution is highly contingent on where we start.
However, all evolutionary biologists say is that natural selection is a proven way for complex adaptations to arise (one of the only ways known, in fact). Combine that with the fact of common ancestry, and it would follow that natural selection provides an excellent explanation for how populations can change and how new adaptations can arise.
Showing that an adaptation has occurred reveals nothing about its mathematically probability of occurrence except that it is nonzero
Technically, if we have shown that an adaptation has occurred reveals that its probability of occurance is 1. Evolutionary biology does not rest on being able to calculate the specific probability of certain types of adaptation. Certain adaptations can be calculated, usually in the form of 'given this locus's mutation rate, it would likely take about 25,000 years of change before species A can have adaption B and species B can have adaptation b.'. This is only practical at the gene level - when we start becoming more general things get less precise.
So, 'what is the probability that gene x will duplicate itself and have an insertion mutation (leading to increased muscle mass)?' Might be, at least in principle, something that can be reasonably calculated.
However, it would be very difficult to calculate what is the probability that this lineage, within two generations, will evolve to have greater muscle mass. Maybe for animals we know well (such as cows), a 'good enough' calculation might be worked out - but there are many ways muscle mass could increase, not just genetic ones either. And of course, there are many ways it could decrease. Some genes may change to lead to an increase, while others change to lead to a decrease.
You said in the OP
It would seem then, that we should examine whether any of the probabilistic arguments are even valid at all, mathematically speaking
On the one hand, evolutionary biologists don't make the kinds of probabilistic arguments you refer to. They are simply gathering evidence and trying to explain it as best as they can. From what we have learned so far, it is known that all life is related and natural selection is one explanation.
We also know quite a bit about biochemistry, enough to be confident that there are certain conditions out there which will begin proto-life.
On the other, its detractors often use very specific probabilistic arguments that 'prove' evolution false - despite the fact that the necessary information to make such calculations is non-existent.
The biologist's answer is not a probabilistically based one, and nor is it claimed to be. In its entirety it is a tentative explanation for what facts we have managed to uncover, with full admission that not everything has yet been satisfactorily explained.
The creationist's answer is probabilistically based. It composes of assuming a non-existent world of randomly bumping particles with no forces but an even and constant temperature that is never revealed and concludes that such a world cannot create life through random bumping alone. It is true, but the conclusion given is usually accompanied by a hidden premise that we live in such a world, and thus life in this world could not have a non-theistic origin. So while it is mathematically 'valid', it is logically absurd.
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