Shortest paths when randomly scrambling a Rubik's cube
At God's number is 20 site you can find a nice distance/count table:



"... Distance-20 positions are both rare and plentiful; they are rarer than one in a billion positions, yet there are probably more than one hundred million such positions. We do not yet know exactly how many there are. The table on the right gives the count of positions at each distance; for distances 16 and greater, the number given is just an estimate. Our research has confirmed the prior results for entries 0 through 14 below, and the entry for 15 is a new result, which has since been independently confirmed by another researcher ...."

According to the table if you pick a random configuration it will be with high probability at distance 17-18 from the solution.

See also D. Joyner's notes Mathematics of the Rubik's cube (and D. Joyner's book "Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys")