I use the Birthday Paradox when I'm searching for matching socks in the morning -- rather than pick a sock and look for one that matches it, I accumulate socks and choose the first pair that match.
My other favorite programming metaphor is about skiplists: they're like a set of subway lines, with some extra express lines that skip a random (and increasing, for each level) number of stops.
It was more about the problem of matching socks when I put them away, but the same principle applies.