Dice are commonly rolled in Dungeons and Dragons to decide the outcome of events. One such case is in combat. First, you roll 1d20 (one 20-sided die) to determine wether your attack hits (attack roll). The attack hits if the value of the attack roll is $ge n$, with $n$ depending on the stats of you and your opponent. Then you do a damage roll to determine the damage dealt by your weapon, with the dice you roll depending on the weapon. The question is, how does the average damage dealt depend on $n$? What if I have two weapons, where I would need to roll 2d20, to see if either hit?
- Roll 1d20 per weapon (1 or 2)
- If the value of the roll is $ge n$, the weapon hits
- For each weapon that hits, roll 1d$m$, where $m$ depends on the weapon
- Deal damage equal to the value of that roll
For a weapon dealing 1d$m$ damage, what is the average damage dealt when the attack roll needs to be $ge n$? What if you have two weapons? Assume both weapons have the same $m$ value.
I know the raw average of a $p$d$q$ roll is $p*(q+1)/2$ (normally distributed), but I don’t know how the first 1d20 roll affects that, given that it effectively causes the damage roll to be 0, thus skewing the distribution.