Too Soon for Doom Gloom?

The observation that we are among the first 10^{11} or so humans reduces the prior probability that we find ourselves in a species whose total lifetime number of individuals is much higher, according to arguments of Carter, Leslie, Nielsen, and Gott. However, if we instead start with a prior probability that a history has a total lifetime number which is very large, without assuming that we are in such a history, this more basic probability is not reduced by the observation of how early in history we exist.