Let n be an integer...You need a prior distribution on n. Without a prior, these probabilities are not 50%. They are undefined.

Who said anything about not having a prior distribution? "Let n be a [randomly selected] integer" isn't even a meaningful statement without one!

What gave you the impression that I thought probabilities could be assigned to non-hypotheses?

Finally, on an empirical level, it seems like there are more false n-bit statements than true n-bit statements.

This is irrelevant: once you have made an observation like this, you are no longer in a state of total ignorance.