Is there any (uncontroversial) theory that rigorously defines what a 50 % probability for heads and tails means? It certainly doesn't mean that in the long run you will obtain the same number of heads and tails because there is a (vanishing) chance that you will always get heads even though the coin is actually fair. And just saying that you will obtain the same or at least similar number of heads and tails with high probability is a circular definition at best.
Does Bayesian thinking just deny the existence of intrinsic probabilities or considers them out of scope? Assuming there is such a thing as true randomness, for example in quantum physics (and it is not due to our ignorance of hidden variables like in Bohemian mechanics), could a Bayesian assign probabilities to the outcomes? Would a Bayesian be mislead into assigning something other than 50/50 for spin up and spin down if he, by chance, only observes spin up although he is really confident that this is just a statistical fluke? If he is mislead, does that mean that uncertainty about the truth of his assumptions creeps in in case the system momentarily deviates from the expected behavior?
Does Bayesian thinking just deny the existence of intrinsic probabilities or considers them out of scope? Assuming there is such a thing as true randomness, for example in quantum physics (and it is not due to our ignorance of hidden variables like in Bohemian mechanics), could a Bayesian assign probabilities to the outcomes? Would a Bayesian be mislead into assigning something other than 50/50 for spin up and spin down if he, by chance, only observes spin up although he is really confident that this is just a statistical fluke? If he is mislead, does that mean that uncertainty about the truth of his assumptions creeps in in case the system momentarily deviates from the expected behavior?