For example, if the population stays at 8B between now and infinity, and we assume a uniform distribution of % chance you're born at any time, then it's just as equally likely that you'd be born at the beginning as at the end. There's no higher chance that you'd be born in the middle, whatsoever.
Now actually the population is likely to grow, so if anything it's much more likely that you'd be born towards the end. That begins said, some people have to be born first and of course if they thought this way, they might say things like "oh what a coincidence, this is highly unlikely" or "It's far more likely due to exponential growth that any one person exist towards the end of humanity, so therefore this must be the end", but of course, this thought could have been had by cavemen 1000's of years ago and they'd be wrong.