User:Yodaj007/Sandbox

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 $h$

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:

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

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

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.

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

Damage

The expected damage from hitting is defined as $d_{hit}=b\,c_{hit}$.

The expected damage from critting is defined as $d_{crit}=b\,m\,c_{crit}$.

The expected total damage from an attack is defined as:

${\begin{aligned}d_{t}&=d_{h}+d_{c}\\&=b\,c_{hit}+b\,m\,c_{crit}\\\end{aligned}}$