Quantum probabilities are not Bayesian, determined from expectations adjusted by experience. Instead they come from the wavefunctions of particles, by the Born rule. The Stern-Gerlach experiment is perhaps the clearest example of the distinct impact of their non-classical qualities.
The Schroedinger's cat arrangement produces 50% likelihood based on a single atom of a radionuclide, and a timespan passing equal to the atoms half-life. Half-lives were initially determined by experiment, but can now (nearly always) be predicted accurately from theory. They decay truly randomly, 'god playing dice', but with a specific distribution related to available decay modes, barrier potential etc. The decay of an individual atom is 50% likely after one half-life, and approaches certainty as time goes to infinity, but might never decay.
We don't know if protons decay, just over a very long time. That's the closest example I can think of to your question. Neutrino observatories additionally look for evidence of proton decays, progressively constraining the minimum proton half-life as they get bigger, and time passes.
As for how many cases lead to a given conclusion, this is well established in science by statistical hypothesis testing. You compare the model with the results, and state something has been discovered when an appropriate level of confidence is reached that the prediction could not have been chance. That level depends a bit: 3 sigma means 99.7% sure of the result, which is fine for most regular physics results; the Higgs-boson was announced at only 2.3 sigma, 98% confidence, but it fitted the models, and as CERN runs the confidence gets higher, over 5 now; gravity waves were all about minimising noise, & they were measured at 5. 1sigma, the 'gold standard' of 5 sigma is already 99.99994% sure; a faster-than-light neutrino result was announced with 6 sigma confidence, but considered so unlikely it was still written off as experimental error.
You get Type 1 & Type 2 errors (and some other higher number errors more rarely used), which specifically relate to the equivalent of 'unlikely coinflips', you might phrase it. Numbers of particle collisions at CERN per second, or atoms in a visible piece of nuclide, are very large - a mole of atoms is enough for a coin flip every second for longer than the current age of the universe. The chances of getting all heads by chance over those timescales, is vastly different to our ordinary experience of coins.
That still leaves examples like the tachyon result: systematic error, bad models, confused thinking. So we like replication of results. Detailed methodologies. The team did in fact, find flaws in their equipment set-up that caused errors far outside their original confidence interval, a fiber optic cable attached improperly, and a clock oscillator ticking too fast.
