That just doesn't seem like a problem to me.
Of course it is. Others are content to point out that your model still includes too many degrees of freedom. The interval on which you've conditioned your likelihood is well enough defined, but the other parameter is not. And the probability distribution function that operates inside your likelihood still needs that parameter, because it's not a trivial distribution for all values of the parameter.
If you define and fix that parameter, we can certainly provide you with a single-valued probability. But then you have to answer my objection, which is why that probability matters? You're way too hung up on the rudimentary algebra of this problem, and you don't consider sufficiently what these models mean. Your inability to reason properly about the probative value of a probability is one of those fatal flaws you admit you can't address. This is why you can't address it.
Is the present time between 1942 and 2042 Gregorian? Obviously, it is -- and, it will be for the next 23 years.
And what about someone reading this thread in 2050? For him, now will
not be in that interval. No, it depends on what you mean by "present time." If I say "present time" is a 500-year interval starting at the instant you read this, then no, the present time exceeds the interval 1942-2042 by a wide margin. And that may be a perfectly valid definition, since this discussion often includes vast swathes of time. I see no reason why "present time" can't be the entire recorded history of humanity, since you're thinking in terms of the vastly larger passage of time since the Big Bang. 10,000 years is not very big compared to 14 billion years. That's my definition of "present time" for the purposes of answering your question.
If, as from the perspective of our potential future reader, "now" or "the present time" should be construed as a relatively short period of time on or about Feb. 16, 2018 (what you evidently intend), then he will want to know what's so bloody special about that date, other than it being the date when the post was written. It has no other significance that matters to him, and no other objective significance. It ends up being a contrived problem with no probative value or interest.
Again, this is you just trying to foist the notion of subjective belief as an operative probability or likelihood. There is no argument that makes solipsism universal and objective so that it works as a mathematical proof. In fact that's pretty much what solipsism is all about.