User:Yodaj007/Sandbox

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1 Definitions
2 Chance of hitting
3 Chance of a critical hit
4 Damage

[edit] Definitions

a: The target AC
r: Roll required to confirm critical hit (low end of weapon critical range)
b: Bonus damage
m: Critical multiplier
h: Bonus to hit, including base attack bonus and all applicable modifiers
p: Amount of power attack taken from $h$

[edit] Chance of hitting

The roll required to hit is the AC of the target minus the players hit bonus, or $\,R=a-h$ . If R is greater than 20, the chance to hit normally 0%. But, it's still possible to hit on a roll of 20, if the player can hit the AC with an assumed roll of 30. If R is less than 1, then any roll other than a 1 will hit. If the required roll is 19, then two values on the die can hit. If it is 15, then 6 values can hit. This can be expressed as:

$\begin{align} c_{hit} &= \frac{1}{20} \begin{cases} 0 & \mbox{if } R > 30 \\ 1 & \mbox{if } 30 \ge R > 20 \\ 21 - R & \mbox{if } 20 \ge R > 1 \\ 19 & \mbox{if } 1 \ge R \end{cases} \\ &= \frac{1}{20} \begin{cases} 0 & \mbox{if } a - h > 30 \\ 1 & \mbox{if } 30 \ge a-h > 20 \\ 21 - a+h & \mbox{if } 20 \ge a-h > 1 \\ 19 & \mbox{if } 1 \ge a-h \end{cases} \end{align}$

$\,c_{hit}$ exists on the interval $\left [0.0,\,0.95 \right ]$ .

[edit] Chance of a critical hit

In order to get a critical hit, you first have to hit with a die roll greater than or equal to r. To confirm the critical, you just have to hit.

$\begin{align} c_{poss\,crit} &= \frac{1}{20} \begin{cases} 0 & \mbox{if } a - h > 30 \\ 1 & \mbox{if } 30 \ge a-h > 20 \\ 21 - a+h & \mbox{if } 20 \ge a-h \ge r \\ 21 - r & \mbox{if } 20 \ge r > a-h \\ \end{cases} \\ c_{conf\,crit} &= c_{hit} \\ c_{crit} &= \left(c_{poss\,crit}\right)\,\left(c_{conf\,crit}\right) \\ c_{crit} &= \left( \frac{1}{20} \begin{cases} 0 & \mbox{if } a - h > 30 \\ 1 & \mbox{if } 30 \ge a-h > 20 \\ 21 - a+h & \mbox{if } 20 \ge a-h \ge r \\ 21 - r & \mbox{if } 20 \ge r > a-h \\ \end{cases} \right) \left(c_{hit}\right) \end{align}$