"If you did 43252003274489856000 random turns on a cube, assuming you start from a scrambled state, then the chance of seeing the solved state by luck is
which is a close approximation of lim n->infinity 1-[(n-1)/n]^n
If you work out this limit you get 1-(1/e) which is approximately a 63% chance.
I know this is probably getting a little too deep into probabilities and stuff, but I just think it's cool that even if you did the exact same number of moves as there are combinations to the cube, there's still only a 63% that your cube will solve by luck."
I think the numbers speak for themselves. You really need to have a system to reach the solved state!