The key insight is that you are unlikely to be experiencing the thing at a special time in its life. This is the Copernican principle (which J. Richard Gott uses in his version of this that Wikipedia mentions), which was basically "we (on Earth) are unlikely to occupy a special place in the solar system -- it's much more likely that some other object is the center."

Gott says you can be 95% confident that you're experiencing the thing in the middle 95% of its life. Let's say x is its life so far. If x is 2.5% of its eventual life (one extreme of the middle 95%), then the thing still has 39x to go. If you're at 97.5% (the other extreme), then the thing only has x/39 left. So the 95% confidence interval is between x/39 and 39x

Of course, 5% of the time you actually are experiencing something at the very beginning or very end of its life (outside the middle 95%), which is a unique thing. But that's why it's a confidence interval < 100% :)

I prefer this form of the principle a lot more than "the expected life is equal to 2X, always."

Side note: I took J. Richard Gott's class in college called The Universe. Maybe not the best use of a credit in hindsight, but we studied some really interesting things like this.