It was a beautiful moment to watch...
Quick Claw (Bulbapedia)
Gen III onwards
Quick Claw gives the holder a 20% chance of going first in its priority bracket (regardless of Trick Room and Stall).
Thus: 0.2^3 = 0.008 0.08%
Gen II
Quick Claw gives the holder a ~23% (60/256) chance of going first in its priority bracket.
Thus: (60/256)^3 = 3375/262144 Roughly 0.13%