I think that the most likely solution to the Fermi Paradox is that, while life is exceedingly common in the universe, intelligent life is incredibly rare. In the four and a half billion year lifespan of this planet, life has existed on it from almost the very first moments. However, not until the Cambrian explosion about 530 million years ago did complex multicellular life exist. That means that for nearly four billion years, and starting from nine billion years after the birth of the universe, Earth contained nothing but single-celled organisms and colonies of such. And it is not until about 2 million years ago, 0.04% of the history of life on Earth, that the first technological intelligence emerged. And even that was simply monkeys hitting rocks against each other in a clever way! Anything more complicated than a stone arrowhead was invented in the last ten thousand years!
And let's look at how unlikely it is that humans (or any other hominid) ever even achieved technology. We had access to wheat, barley, and rice, easily domesticable plants that produced high yields and had good nutritional content. We had access to large animals with very particular internal dominance hierarchies which were not so recently evolved alongside us to attack us on sight, but not so distantly separate from us to be completely unwary at the sight of hunters with spears. (Here by "we" I am referring to any subset of humans who had such access — Jared Diamond goes into great detail as to who in fact had access to what.) We had access to workable stone, copious woodlands, and various ore deposits. We had an abundance of fresh water on a planet with an atmosphere suitable for lighting fires. Our luck in the development of our culture and eventual civilization was astounding. We were never subject to an extinction-level impact or eruption (although it seems that we came damn close).
I think that such luck is not only astounding, but is in fact astronomical. While life seems to have no trouble at all finding a place on a planet like the Earth (and perhaps on many other types of planets as well), technological civilization seems like and absolutely ludicrously unlikely event. It has happened exactly once in four and a half billion years (for 0.0002% of that time), or in 530 million years in which complex life has existed on Earth (or 0.0019% of that time). So I would imagine that life is in fact very common in the universe, although almost exclusively in the form of single-celled creatures living on rocks and in oceans. Very, very, very rarely one would find a planet with some sort of multicellular life on it — simple plant-like creatures, or molds of some sort. And once in an unimaginably huge while, one might expect to find a planet where technologically intelligent life once existed. Finding a planet where technologically intelligent life exists concurrently with us seems depressingly close to a fantasy.
[Edit: My percentages were two orders of magnitude off! They were simple ratios, not percentages. Thanks to Scott for pointing this out.]
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