To begin with, we have to be careful what we mean by "random." Clearly pi is not "random" in the strict sense, because individual digits are certainly not random but mathematically fixed. Perhaps a better and easier question is whether pi is "normal base 10."
The fascination with π is so universal that in 1998 film director Darren Aronofsky even created a psychological thriller with that title. Here I want to tell only a few more π stories that are perhaps less known outside the circle of true aficionados.