...
It's just that the odds of that being the case are very, very, very (rinse and repeat a lot) low. Of course the odds are exactly the same for your point along the timeline, or for any other point along the timeline within the bounds as proposed, to the point where statistically speaking it doesn't make sense to even try.
Yep, you're right of course. After 667e27 years the chance of finding the right k is 100%. In statistics the expected value or the mean would be halve of that, 333.5e27 years. But the odds or so low, it could be zero in a human life time for that matter.
Off topic (because the topic is about very big and very small numbers, and I think it's mind boggling):
1 + 2 + 3 + 4 + 5 + ... = -1/12
1 