Penetration (PV)

This article has information that is missing or not up to par.
Reason: waiting for math extention to make this page look pretty. What is a critical hit's role in all of this? TODO: add probability calculation table for quick reference

PV or Penetration Value or → is a value assigned to all weapons and plays a key part in damage calculation during attacking.

Melee Weapon PV Calculation

Melee weapons have a base PV of 4. The PV will increase by the creatures Strength modifier with an upper limit dictated by the weapon's bonus cap. Two handed weapons and the Sharp mod also increase base PV by 1.

A critical hit(natural 10) is guaranteed one penetration.

Ranged Weapon PV Calculation

This article has information that is missing or not up to par.
Reason: is this the same for grenade launchers, thrown weapons, or bows?

Ranged weapon PV is determined by the PV of the gun itself. The PV of the bullet does not matter.

Critical hits on ranged weapons add a flat +4 PV to the shot.

Penetration Formula

For a round of penetration, 1d10 is rolled three times.

$PenetrationRoll=(1d10+PV-((PenetrationRound)\times 2))$

If any of these + the attackers PV are greater than the opponents AV, the attack will penetrate once. If all three 1d10s + PV are greater than the AV, the attack penetrates once and another round of penetration is done, rerolling 3 more 1d10s + PV-2.

The chance of $n$ penetrations goes by the formula:

$3\left(1-{\frac {AV-(PV-2n)}{10}}\right)-3\left(1-{\frac {AV-(PV-2n)}{10}}\right)^{2}+\left(1-{\frac {AV-(PV-2n)}{10}}\right)^{3}\times \left({\frac {AV-(PV-2(n+1))}{10}}\right)^{3}$

If PV = AV, the formula simplifies to

$3\left(1-{\frac {2n}{10}}\right)-3\left(1-{\frac {2n}{10}}\right)^{2}+\left(1-{\frac {2n}{10}}\right)^{3}\times \left({\frac {2(n+1)}{10}}\right)^{3}$

The probability table would be

Penetrations	Probability distribution	Probability (cumulative)
0	0.008	0.008
1	0.512768	0.520768
2	0.392527872	0.913295872
3	0.752768	0.996545991
4	0.488	1.000000
5	0	1

Derivation

${\frac {AV-(PV-2n)}{10}}$ is the probability that a single roll at $n$ rounds of penetration fails.

If $A_{n}$, $B_{n}$, $C_{n}$, are each the probability 1 successful roll at n rounds of penetration:

$\rightarrow A_{n}=B_{n}=C_{n}=1-{\frac {AV-(PV-2n)}{10}}$

The probability of n penetrations becomes a sum of the chance that that 1 or 2 out of 3 rolls hit:

$(A_{n}\lor B_{n}\lor C_{n})-{\big (}(A_{n}\land B_{n})\lor (B_{n}\land C_{n})\lor (A_{n}\land C_{n}){\big )}$

OR the probability that all 3 rolls hit, but the second round of penetration fails to penetrate at all:

$(A_{n}\land B_{n}\land C_{n})\land (\sim A_{n+1}\land \sim B_{n+1}\land \sim C_{n+1})$