Good point, I like this one.

To retranslate it into the urn/ball thougt experiment:

"Imagine that two big urns are put in front of you, each containing a million red and blue balls. You know that one of them contains red and blue balls fifty-fifty and the other a thousand red balls in just under a million blue ones, but you are ignorant as to which is which. Now you take a ball at random from the left urn, and it turns out to be red. Clearly, this is a strong indication that that urn contains red and blue balls fifty-fifty.

Now consider the balls depicting millenia, where red balls stand for the time humanity exists, it is easiy to see that humankind must exist for a seizable portion of the lifespan of the universe."

But as I am writing this, I see a catch: Arguing from Carter's perspective, it is now obvious that doom is imminent not only for mankind, but for the universe as a whole: The fewer blue balls are in the second urn (i.e. the shorter the livespan of the universe), the better is the probability to draw a red one. Hence, the probability of being alive today is maximized if the universe ends tomorrow.

I am starting to like this line of reasoning. I guess one can prove everything (and its opposite) applying probabilities "ex post" on preexisting facts...