Sorry, I missed this one on the first run through.
Not at all. If a small population has existed for a small period of time, only a few people have ever lived, and Carter's reasoning predicts that that population is likely to die out after only a few more people have lived. A "small disaster" might be sufficient to achieve that, given that the population is small. If a large population has survived for a very long time, then Carter predicts a large number of future lives before that population dies out. Probably a "large disaster" would be required. Since small disasters are more frequent than large disasters, this is all internally consistent: frequent small disaster eliminating small, short lived populations, and occasional large disasters wiping out the infrequent large populations who've been lucky enough to slip through the "small disaster" winnowing. (There's certainly genetic evidence that humans have suffered a near-extinction event when our population was small: we're genetically much more similar than we should be, given the length of time we've been around on the planet as a species.)

I think the exponential growth of the human population is what makes Carter's argument counterintuitive to some folk. If the human population were stable, then Carter's reasoning would reduce to:
"Hey, we've been around for a couple of million years without being wiped out. Chances are we can survive another couple of million. It's a 50:50 chance whether we last for a longer or shorter time than that."
Probably most people would find this an unexceptional bit of informal reasoning.
But Carter's calculation applies to human lives, rather than elapsed time, and if the distribution of human lives is skewed rightwards along the time axis, giving us a long pastward tail and a short futureward tail, people begin to become uneasy.

Well, it's been fun. I'm about to disappear into the Scottish mountains for a few days. I'll try to pick up this thread when I get back (assuming we're all still here).

Grant Hutchison