This is first assuming that you can take the middle pieces out. There are 6! different combinations of how to put on the centers from one angle. But many are the same but just from another angle. This reduces it by a factor of 2(6), so the number drops to 60.
There are 8! different places to put the corners on, and that is multiplied by the flip coefficient (or whatever) of 3 different flips and 8 places the flips occur, so 3^8. That gives us 8! x 3^8. But as before, there are 24 clones per the actual number, so it becomes (8! x 3^8)/24, or 7! x 3^7.
There are 12! different places to place the edges, with 2 flips, so it becomes 12! x 2^12. But once again, there are clones in that calculation. There are 24different places to use as your point of view when looking at a cube (Don't say it's only six, because you can rotate any face 4 times), so it's reduced by a factor of 24. So the number becomes (12! x 2 ^ 12)/24, or 11! x 2^11.
Finally, it's time to multiply 60 (centers) by 11,022,480 (corners) by 81,749,606,400 (edges). If you divide it by the 4.3 quintillion possibilities, this yields a "1 in 9.6 chance" or 10.4%.