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A couple of questions on Counter/Mirror Coat?

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If the Pokémon using Counter is hit by a physical attack that deals 0
damage, Counter becomes a physical move with 1 base power.

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1) What is the chance of a move(s) doing zero damage?
2) What moves do zero damage?

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Where did you get this from?
bulbapedia
Well look at this way, if a Shedinja was hacked to have Counter and it was hit with a super effective physical move whilst holding a Focus Sash, it wouldn't lose any HP because it only has 1 HP to begin with. Because the physical attack dealt 0 damage and because Counter counters all physical moves for twice the damage done to the user, Counter would become a physical move with 1 Base Power, even though double 0 is... well... 0. This is because the physical attack used to deal 0 damage is still affected by Counter and because Counter can't really deal 0 damage without failing.

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Best answer

Hold Back and False Swipe can deal 0 damage, as they always leave the target with 1 HP.

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and mirror coat?
Both Hold Back and False Swipe are physical moves, but it is resonable to assume Mirror Coat would also deal 1 damage if a special move could deal 0 damage.
Just to clarify, there is no move that deals 0 damage other than those two moves. Other moves would either do 1 damage or they'd simply fail.
There's no "chance" of dealing 0 damage, because the damage formula is set up so moves always deal at least 1 damage. The only exception to that is those two moves, which deal 0 damage if they hit a target with 1 HP.