Most of us wouldn't have come up with the Doomsday Argument.

Therefore, we are not a random selection of the human race. We are part of the subset of the human race that existed after the Doomsday Argument was raised.

Thus the initial assumption of the argument is incorrect.

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"I worry that, especially as the Millennium edges nearer, pseudo-science and superstition will seem year by year more tempting, the siren song of unreason more sonorous and attractive." - Carl Sagan, 1995