For example, we have a tendency to infer non-randomness from apparent patterns in random events (witness the incorrigible optimists who spot trends in the spins of a roulette wheel or the ups and downs of the FT Share Index); at the same time, the history of statistics suggests that, when random samples are required, we often mistake the merely haphazard - or whatever happens to be near at hand - for the truly random.
In my continuing attempts to gauge my audience, I would guess that your eyes would glaze over were you to attempt to read the paper, or this, which is a pretty good explication of the Doomsday Argument, and which presents it in relation to the weak anthropic principle. It has been proved, after all, that you are Humanities-studying iPod owners (I mean, okay, I imagine Sam probably spent some hours of class time discussing the Doomsday Argument, but the rest of you probably not so much).
What's interesting about The Doomsday Argument (and if you haven't clicked one of the links, the explanation I gave previously, while not quite the same, works fine: it seems like you'd be statistically more likely to be born in the latter 2/3rds of all humans who have ever been born than in the former 1/3, but if that were the case then humans would have to become extinct in a few hundred years) is that at first it's hard to understand why this is actually any sort of philosophical problem--it just seems like The Damned Lies of Statistics (which, as I will one day blog about, are not Damned Lies of Statistics, they are Damned Lies of Language. Many people are just calling them "lies" these days). Then after you've groked the argument, it's equally hard to understand why your initial arguments against it don't quite work--it's based on the idea that the fact that You are Here, Now has some particular intrinsic meaning--or rather, that it doesn't, that you are a random sample from the grab bag labeled, "all humans in history." This makes counter arguments difficult, because You are, in fact, Here Now (e.g., counter-arguments like, "Yeah, but the Doomsday Argument has been true for everyone who has ever lived or is currently alive," seem like they're putting you on an equal observational footing with every other person in the "experiment." They're not. If there are only every going to be, say, 100 billion humans in the history of time, including humans 1 through 99,999,999,999 in your experiment doesn't give you any more information than you already had. You'd need the 124 billionth human in your sample to get any premise-shattering data, and you can't have him or her (or it). As soon as you state the terms of the argument, you put yourself at the argumentative "end of the line," as it were, and...okay, already your eyes are glazing over. You have no idea what I'm saying right now. I've completely lost you...uh...never mind).
What's also interesting about the Doomsday Argument is that I think problems like this are windows into new models. Whatever our current models for understanding probability and observation and, you know, being itself are, they can't quite handle this problem, which means some tear-down and rebuilding is indicated.
By the way, I stumbled upon all of this via this post, which I found through about Five Degrees of Blog Separation, a phenomenon with which, since my post on audience the other day, I have become obsessed. Interestingly, this guy thinks this paper solves the issue, which I don't at all.