The typical lottery winner is pretty danged mediocre. They just got lucky.
That said, their is a lot to be said for the motion that the Earth is extraordinary. It's the right size and in the right place, it has a big honking Moon, it orbits a star that does not have a binary pair, is relatively stable, is neither too big nor too small, is neither too far from nor too close to the galactic center. Then there's Jupiter, which (1) exists, helping to clear out the junk from the solar system, and (2) didn't go on a kamikazee sunward dive.
Simple life happened to develop on Earth (quickly, so based on a sample size of one, that's not so extraordinary), then complex life happened to develop on Earth (rather late, so so based on a sample size of one, that might be extraordinary). While semi-intelligent life has arisen multiple times, intelligent life in an environment conducive to advancement (how would hyper-intelligent squid develop fire, extract metals, build radios, get out into space?) and equipped with manipulators conducive to advancement has arisen once (based on a sample size of one). In the 4.5 billion years that it took for that one species to develop, the universe has not conspired to wipe out life (it's tried, multiple times), life hasn't committed suicide (it's tried, multiple times), conspired to wipe out that one intelligent species (it's tried, multiple times), and that one intelligent species hasn't committed suicide (yet). Life has gone ballistic a couple of times, leaving us with all those neat hydrocarbons without which a hyper-intelligent version of us would still be playing around with fire.
In short, the Mediocrity Principle may well be yet another pile of hogwash brought to us by the humanities. The Earth might be anything but mediocre. Intelligent life might well be extremely rare.
The Drake equation can be wielded to give answers ranging from "we are utterly alone" to "the universe is utterly crowded": It's pretty much worthless. Until we get beyond the basic problem of extrapolating from a sample size of one, the best answer is we don't know -- yet. The Kepler mission is a first small step toward getting past this problem.