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The monk's Deflect Missiles feature description says (PHB, pg. 78):

Starting at 3rd level, you can use your reaction to deflect or catch the missile when you are hit by a ranged weapon attack.

I take this to mean that the player controlling the monk gets to decide whether or not to use Deflect Missiles after they know whether or not the attack hits.

It continues:

When you do so, the damage you take from the attack is reduced by 1d10 + your Dexterity modifier + your monk level.

Does the monk's player get to know the damage before they decide to use Deflect Missiles? Or is damage is rolled after they choose to use the feature?

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You choose before you know the damage. Hitting and dealing damage are separate steps — if the ability was supposed to give the choice when damage was being resolved, it would say something like "… when you are damaged by a ranged weapon attack." Instead, it says "when you are hit". So when you're hit, but before damage is rolled (or revealed), you decide whether to deflect it or not.

Besides, it's a deflection. It wouldn't make sense to see how wounded you are first, and only then decide whether to make the missile hit you differently or not at all. The ability's use of a hit as the trigger instead of suffering damage as the trigger is in line with this intuition about how voluntary deflection ought to work.

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    \$\begingroup\$ I agree with this interp but would add a caveat that a generous DM could hint at the damage by suggesting where the missile MIGHT hit (center chest [high damage] or wing an arm [low damage]) and allow the player the choice. Plus the missile could be magicked or poisoned so even a scratch could have bad effects (assuming the DM rules that if damage is reduced to 0 the missile doesn't hit). \$\endgroup\$ – Jason K Jan 6 '15 at 16:23
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Strictly speaking, if you know the damage you got hit with, then you have already been damaged and it's too late to reduce it. This isn't a heal reaction or ability. You are reacting to an incoming missile not a missile that has already hit. It is assumed that every character not incapacitated, stunned, or otherwise unable to react is trying to avoid being hit at all times during a fight. That is why we all have a Dex modifier to our AC whether a monk or not, and why we have a Dodge action.

A monk, however, has a little bit of extra kung fu-ey style defense with flair. If the attack roll ends up being a hit, the Deflect Missiles feature gives you a second chance to avoid damage and sometimes avoid being hit at all. That is, "I didn't quite dodge it, but I was able to stop a lot of the damage by focusing my Ki and knocking it away with my hand before it penetrates fully." If it's a good enough deflection (damage reduced to zero as a mechanic), then the missile was caught cleanly in the hand and can be thrown as an attack using the monk's reaction.

I know there will be a lot of room for interpretation by various DMs, but I think this is both the RAW and the RAI point of view.

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