Consider for a moment a simple line of reasoning. What is the probability that you would be born in this generation rather than a previous or future generation? Well, to figure this out we’d want ideally to know the total number of human beings who have, are, and will ever live. Then we would use this number to divide the number of people living in any given generation to find the distribution. That’s not particularly knowable at this point.
However, we do know the rough populations of generations preceding the current one, as well as the current one (http://www.prb.org/Articles/2002/HowManyPeopleHaveEverLivedonEarth.aspx). We thus know that there has been a steady increase in population over time. Given this trend, it would be simple enough then to assume that you’re more likely to be born in the future than in the past, because there are always more people in the future due to population growth.
This however does not take into account the possibility of eventual human extinction. That has two potential outcomes, a gradual decline, or a sudden dramatic drop. In the gradual decline scenario we could possibly assume that the human population follows some kind of bell-shaped curve (though the fact of exponential growth confounds this idea somewhat). Alternatively, the highest point precedes a sudden decline (think, thermonuclear war). In either case, the law of averages suggests that since you were born in this particular generation, that this generation is approaching the apex of human population growth.
We can also apply to this, parallel universes. There may be an infinite number of possible worlds, but the one in which you are most likely to be born in is this one. That suggests that in a large number of possible universes, humanity’s population drops after this point in history. This suggests that universes in which humanity populates the stars in massive numbers are rare.