PokéBase - Pokémon Q&A
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It was a beautiful moment to watch...

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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%

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Oh wow that's cool! Thanks for answering!
Happy to help!