No. If the the total population were eleven, you and I would be 100% guaranteed to be among them. If the population were 50 billion, you and I would be 100% guaranteed to be among them. (If the population were one, I would be 100% guaranteed to be it, and I'd be talking to a figment of my own imagination, or someone who disintegrated while I was typing this.)
Since we
do exist, our probability of having been born is exactly 100%. The same as everyone else who exists.
The probability of
more people being born grows with the population (all else being equal - in actual fact, the birth rate does fluctuate with various environmental and cultural influences); the probability of each individual remains the same: 100%, even while each individual becomes a smaller percent of the total population.
I did define this scenario as there being an end to the human species so we know n births.
I don't understand that part.
I see that there is a physical limit to growth: either we stop overreproducing or we run out of resources and die. (Again, all other things being equal: the asteroid passes at a safe distance; the volcanoes take turns erupting, etc.) If that limit is to be reached within the next 50 years, you have a far greater chance of seeing the end than I do: your life expectancy is longer.