Why are knots so common? This article in Science News, which I found from Slashdot, gives a bit of an explanation. (There are links to further, more scholarly, articles in the Science News piece; it's Sunday morning, though, so I'm not bothering to read them.)
My instinct is that any measure of knottedness is bounded below, but not above. You can't get more unknotted than, well, an unknotted string, but conventional measures of knottedness (say, the crossing number) can get arbitrarily large. So we can imagine a random walk on the positive half-line, which takes values which are the crossing numbers of some random process which generates knots; the crossing number will tend to be some medium-sized positive number, just because whenever it's zero it gets forced away.