If with every birth there is some chance c that I specifically will be born, then the number of total people born (N) does not effect the odds that I will be in the first 100 billion people. My odds of being in the first 100 billion people would be 100,000,000,000*c which is unaffected by N. In a 100 trillion person universe my odds of being in the first 100 billion would be lower than being somewhere closer to the middle, but in a 200 billion person universe my odds of being born at all are much lower. Is there a flaw somewhere in this reasoning? Does the doomsday argument assume that you will be born?
You are already born. This is a fact, knowledge you have. You know you are in the first 100 billion people, because that is approximately the number estimated for total humans born until now.
The doomsday argument argues that, given that:
- there is a finite number of humans that have been and will ever be born
- principle of indifference, you are equally likely to be the x'th person as the x+1'th
With these, you can argue that with a 95% probability, you live among the last 95% of humans, so you can get an upper bound estimation on how many humans in total there will ever be.
This is important. We are interested in an upper bound. Not that we are also 95% likely to live among the first 95%, giving us a lower bound estimate on how long humanity will survive at least. But that's not what we are interested in.
You are not more likely to be in the middle than at the beginning or the end of the distribution, you are equally likely.