I have an old 12-colored Megaminx that I put all new stickers on because the old ones were falling off. This Megaminx was in more of a state of disrepair than I originally thought, though, and when I was solving it 2 of the pieces (1 edge and 1 corner) popped out and fell on the floor. I wasn't paying attention to those particular pieces, so I had no idea which way they were facing when they popped out.
I plugged them back into the puzzle. I had no idea if they had the correct orientation or not though. Surprisingly, I was still able to complete the puzzle without disassembling it.
I know that a Rubik's Cube has parity; only $\frac{1}{12}$ of the ways to assemble its cubelets results in solvable cubes. My intuition tells me that the Megaminx obeys the same parity rules (when I first got my Megaminx back in high school I was able to solve it using only the algorithms that come with a Rubik's Cube, with a few minor tweaks); however, I lack the mathematics background to verify this.
My question: If I were to completely disassemble a Megaminx and reassemble it at random, what are the odds that the resulting state would be solvable?