The only way for anyone to learn the correct law would be to run experiments on themselves and observe the frequencies.

That's what I was going to propose, after Emile said he doesn't know what the probabilities mean.

If we adopt a frequentist interpretation, then I suppose the probability makes sense. After all, this is the only way we can actually observe the Born probabilities in quantum mechanics: by repeating the experiment and counting frequencies.

But.

Even if I ran the experiment on myself, what would make my subjective report any more valid than that of the other copies? In quantum mechanics, even assuming MWI, the opinions of our copies in the other branches are inaccessible to us. Suppose I copied myself 10 times. Then the resulting 1024 copies will have every possible subjective experience generated among them. I guess we could say that what the 50% probability really means is that the resulting copies would obey the same binomial distribution that you would have after generating every possible sequence of 10 coin flips.