Let n be an integer. Knowing nothing else about n, would you assign 50% probability to n being odd? To n being positive? To n being greater than 3? You see how fast you get into trouble.

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

The particular mathematical problem is that you can't define a uniform distribution over an unbounded domain. This doesn't apply to the biased coin: in that case, you know the bias is somewhere between 0 and 1, and for every distribution that favors heads, there's one that favors tails, so you can actually perform the integration.

Finally, on an empirical level, it seems like there are more false n-bit statements than true n-bit statements. Like, if you took the first N Godel numbers, I'd expect more falsehoods than truths. Similarly for statements like "Obama is the 44th president": so many ways to go wrong, just a few ways to go right.

Edit: that last paragraph isn't right. For every true proposition, there's a false one of equal complexity.