Penetration (PV): Difference between revisions

Penetration (PV) (view source)

Revision as of 20:28, 8 July 2019

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→‎Penetration Formula: latexed the penetration for safe and fun

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 For a round of penetration, 1d10 is rolled three times.
-penetration = (1d10 + PV-((penetration round)*2))
+<math alt="Penetration Roll equals 1 d 10 plus PV minus two times Penetration Round">PenetrationRoll = (1d10 + PV-((Penetration Round)\times 2))</math>
 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:
+The chance of <math alt="n">n</math> penetrations goes by the formula:
-(1-(AV-(PV-2n))/10) - 3(1-(AV-(PV-2n))/10)<sup>2</sup> + (1-((AV-(PV-2n))/10)<sup>3</sup>*(AV-(PV-2(n-1))/10)<sup>3</sup>
+<math>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</math>
-which is derived from:
+If PV = AV, the formula simplifies to
-If A<sub>n</sub>, B<sub>n</sub>, C<sub>n</sub>, are each 1 successful roll at n rounds of penetration:
+<math>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</math>
-(A<sub>n</sub> v B<sub>n</sub> v C<sub>n</sub>) - (A<sub>n</sub> ^ B<sub>n</sub> ^ C<sub>n</sub>) (the probability that 1 or 2 out of 3 rolls hit)
+The probability table would be
-OR
+{|
+! 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
+|}
-(A<sub>n</sub> ^ B<sub>n</sub> ^ C<sub>n</sub>) ^ (~A<sub>n+1</sub> ^ ~B<sub>n+1</sub> ^ ~C<sub>n+1</sub>) (the probability that all 3 rolls hit, but the second round of penetration fails to penetrate at all)
+===Derivation===
+<math>\frac{AV-(PV-2n)}{10}</math> is the probability that a single roll at <math alt="n">n</math> rounds of penetration fails.
+If <math alt="A sub n">A_n</math>, <math alt="B sub N">B_n</math>, <math alt="C sub N">C_n</math>, are each the probability 1 successful roll at n rounds of penetration:
+<math>\rightarrow A_n=B_n=C_n = 1-\frac{AV-(PV-2n)}{10}</math>
+The probability of n penetrations becomes a sum of the chance that that 1 or 2 out of 3 rolls hit:
+<math>(A_n \or B_n\or C_n) - \big( (A_n \and B_n) \or (B_n \and C_n) \or (A_n \and C_n) \big)</math>
+OR the probability that all 3 rolls hit, but the second round of penetration fails to penetrate at all:
+<math>(A_n \and B_n \and C_n) \and (\sim A_{n+1} \and \sim B_{n+1} \and \sim C_{n+1}) </math>
 [[Category:Mechanics]]