[4E] Ancient Forebearer's Rage, Vorpal, and various weapons

[Gralamin]

So, I was approached with the question "If a Barbarian with a +6 Vorpal Falchion and Ancient Forebearers Rage attacked someone, what would their damage look like?"

As it turns out, the Expected Roll of a 1d4 weapon is 15. The Expected Roll of a 2d4 weapon is then 30.

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Math Proof:
Possible rolls: 3, 7, 11, ... , 4n+3, ..., infinity.

Expected Value = 3 * p(3) + 7 *  p(7) + 11 * p(7) + ... + (4n+3) * p(4n+3) + ... + infinity * p(infinity)
p(3) = 1/4 // Roll a 3
p(7) = 3/4 * 1/4 // roll a 1, 2, or 4 and roll a 3.
p(11) = 3/4 * 3 / 4 * 1/4 // roll a 1, 2, or 4 and 1, 2, or 4, and 3
etc.

Expected Value = sum_{n=0}^{infinity}(4n+3)(3/4)^n(1/4)
= (1/4)sum_{n=0}^{infinity}(4n+3)(3/4)^n
= (1/4){4*sum_{n=0}^{infinity}(n)(3/4)^n + 3*sum_{n=0}^{infinity}(3/4)^n}

This is a taylor series + a geometric series
= (1/4){4 * (3/4)/(1/16) + 3 * 1/(1/4)}
= (1/4){3 * 16 + 3 * 4} = 60/4 = 15

Therefore the expected value is 15.

However, the maths behind other rolls are not easy. So I wrote the following program:

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#include <iostream>
#include <cstdlib>
#include <ctime>
#include <string>

using namespace std;

int VFDX(int x);
int randomInt(int, int);


int main()
{
    unsigned int sum4 = 0;
    unsigned int sum6 = 0;
    unsigned int sum8 = 0;
    unsigned int sum10 = 0;
    unsigned int sum12 = 0;
    double average;
    srand((unsigned)time(0));
    const int TRIALS = 100000000;
    
    for (unsigned int i = 0; i < TRIALS; i++)
    {
        sum4   += VFDX(4);
        sum6   += VFDX(6);
        sum8   += VFDX(8);
        sum10  += VFDX(10);
        sum12  += VFDX(12);
    }
    
    average = (double) sum4 / (double) TRIALS;
    cout << "VFD4 Average: " << average << endl;
    average = (double) sum6 / (double) TRIALS;
    cout << "VFD6 Average: " << average << endl;
    average = (double) sum8 / (double) TRIALS;
    cout << "VFD8 Average: " << average << endl;
    average = (double) sum10 / (double) TRIALS;
    cout << "VFD10 Average: " << average << endl;
    average = (double) sum12 / (double) TRIALS;
    cout << "VFD12 Average: " << average << endl;
    
    system("PAUSE");
    return 0;
}

int VFDX(int x)
{
    int roll = randomInt(1, x);
    if(roll == 1 || roll == 2 || roll == x)
        return x+VFDX(x);
    else
        return roll;
}

int randomInt(int low, int high){
    int random;
    int range;
    
    high += 1;
    low -= 1;
    range = (high - low) + 1;
    
    do {
          random = low + int(range * rand() / (RAND_MAX + 1.0));
        } while (random >= high || random <= low);
    return random;
}

Which gives the following results per die:

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VFD4 = 14.9984
VFD6 = 9.99986
VFD8 = 9.79977
VFD10 = 10.2862
VFD12 =  10.9999

My MathFu is too weak to tell me the exact value of anything >D4 (I was getting 7.4444444 for D12), but I'm sure Yakk or someone could give exact values.

Edit: Full values found!
The Expected damage value for a weapon of Die Size X, where X >= 3

D = 1/X * (sum from 3 to X-1) + 3/X * (D + X)
XD = (X)(X-1)/2 - 1 - 2 + 3D + 3X
2(X-3) D = (X^2+5X-6) = (X+6)(X-1)
D = (X+6)(X-1)/(2X-6)

Using in Damage Calculations
The following are exact values:
{table=head]Die|Value
d4|15
d6|10
d8|9.8
d10|72/7
d12|11[/table]
In a given damage amount, add up all dice of the same type, and then multiply by the corresponding values. Then add all the dice together.

For example, if an attack deals 5d6+3d4+6d12 damage (For some reason), then it would deal 5*10+3*15+6*11 = 50+45+66 = 161 damage on average.

Hope this is useful to someone!

Edit:
Ancient Forebearers without Vorpal

E(d4) = 3*1/4 + 4 * 3 /4 = 3.75
E(d6) = 1/6 * (3 + 4 + 5 + 6 * 3) = 5
E(d8) = 1/8 * (3 + 4 + 5 + 6  + 7 + 8 * 3) = 6.125
E(d10) = 1/10 * (3 + 4 + 5 + 6  + 7 + 8 + 9 + 10 * 3) = 7.2
E(d12) = 1/12 * (3 + 4 + 5 + 6  + 7 + 8 + 9 + 10 + 11 + 12 * 3) = 8.25

The above are exact values.

[Pramxnim]

Awesome! Thanks for doing the math for this, man!

Now there's some definite numbers to support the claim that AF + Vorpal Falchion is overpowered (30 average damage per [W]... gg). There's also a chance that the die will keep rolling and rolling too... it'll never make it into any of my games, that's for sure!

[Gralamin]

Awesome! Thanks for doing the math for this, man!

Now there's some definite numbers to support the claim that AF + Vorpal Falchion is overpowered (30 average damage per [W]... gg). There's also a chance that the die will keep rolling and rolling too... it'll never make it into any of my games, that's for sure!

It can get even worse. Consider if you attack four times against something you have a 55% chance of hitting. Lets say each hit does 2[W]+Strength, each miss does 0. Say you also have a 19-20 crit range.
Say the mods work out as +22 to damage. For each attack, we have a damage of expression of:
MissChance * MissDamage + (HitChance-CritChance)*(HitDamage) + CritChance * CritDamage
= 0.45*0 + 0.45 * (15*2+22) + 0.10 * (15*6 + 6 * 11 + 22 + 16)
= 42.8 damage an attack, on average
So in the example you deal 171.2 damage on average. This is likely a weak example, and you can easily tweak it to get much higher damage amounts.

[Pramxnim]

Actually, the equation above assumes a 1d4 weapon, since you had 15*2 as the average damage per die.
For a 2d4 Falchion, it would be 15*4 = 60 avg damage per 2[W] attack. The average damage would have doubled then.

[Nightson]

AFR-
"Until the rage ends, whenever you roll 1 or 2 on a damage die for a primal attack, the roll changes to the die’s maximum value."

Vorpal-
"Whenever you roll the maximum result on any damage die for this weapon..."

AFR triggering the vorpal weapon property is a pretty sketchy interpretation.

[Pramxnim]

It is, isn't it? But yeah, there is a subtle difference between AF Rage and other abilities that modify your damage roll. If I recall correctly, all of the others say: "Treat your damage as if you had rolled a X", whereas AF Rage says: "Change the roll so that it comes up as maximum value"

Since you rolled the damage roll before AF Rage triggers, you could say that you rolled maximum damage and thus trigger Vorpal. Sane DMs should know of this trick to shut it down before it gets abused (well, as much as a level 29-30 character could abuse it anyways, since the game should be nearing its end at that point).

[Gralamin]

First, yes I did screw up my example while half asleep. Point still stands.

AFR-
"Until the rage ends, whenever you roll 1 or 2 on a damage die for a primal attack, the roll changes to the die’s maximum value."

Vorpal-
"Whenever you roll the maximum result on any damage die for this weapon..."

AFR triggering the vorpal weapon property is a pretty sketchy interpretation.

It is, isn't it? But yeah, there is a subtle difference between AF Rage and other abilities that modify your damage roll. If I recall correctly, all of the others say: "Treat your damage as if you had rolled a X", whereas AF Rage says: "Change the roll so that it comes up as maximum value"

Since you rolled the damage roll before AF Rage triggers, you could say that you rolled maximum damage and thus trigger Vorpal. Sane DMs should know of this trick to shut it down before it gets abused (well, as much as a level 29-30 character could abuse it anyways, since the game should be nearing its end at that point).

Indeed, the Changes part makes it so it works - You roll maximum on anything other then a 3.

[tcrudisi]

It's a level 29 daily -- the highest daily power in the game. At this point, awesome things should happen. Is it broken? Probably not, because we are talking about level 29 or 30 here. If this occurred at level 15? Heck yes. But the characters have almost fulfilled their epic destiny... I would want them to be truly epic.

[Gametime]

I'm of the opinion that Vorpal is an enchant you should never let your players find or buy, but that has more to do with the flavor behind it. If you want a sword shard enough to behead a Jabberwocky, you better be prepared to enchant it yourself.

At any rate, if they can't do it until level 30, it should be fine. Level 30 is silly anyway.

[Krazddndfreek]

You do realize that in 4e Vorpal doesn't do an auto-kill right? Only gives bonus damage rolls on max damage rolls. Exactly like how Aura of Chaos works from Devoted Spirit.

[Artanis]

You do realize that in 4e Vorpal doesn't do an auto-kill right? Only gives bonus damage rolls on max damage rolls. Exactly like how Aura of Chaos works from Devoted Spirit.

That damage is what he's talking about. With Ancient Forebearer's Rage, a 1d4 Vorpal weapon does about 15 damage per [w].

[Sir Homeslice]

Awesome! Thanks for doing the math for this, man!

Now there's some definite numbers to support the claim that AF + Vorpal Falchion is overpowered (30 average damage per [W]... gg). There's also a chance that the die will keep rolling and rolling too... it'll never make it into any of my games, that's for sure!

Oh dear lord, someone is being powerful at level 29. Someone alert the CIA!

Do you think maybe you're overreacting?

[tbarrie]

However, the maths behind other rolls are not easy.

They kind of are, actually. Take a D6: You have a 1/6 probability of doing 3 damage, a 1/6 probability of 4, a 1/6 probability of 5, and a 1/2 probability of 6 plus whatever the expected value is. So if we assign the expected value the label "D", we have:

         D = (1/6)*3 + (1/6)*4 + (1/6)*5 + (1/2)*(D + 6)
         D = (12/6) + (D/2) + 3
 D - (D/2) = 2 + 3
       D/2 = 5
         D = 10

[Gralamin]

They kind of are, actually. Take a D6: You have a 1/6 probability of doing 3 damage, a 1/6 probability of 4, a 1/6 probability of 5, and a 1/2 probability of 6 plus whatever the expected value is. So if we assign the expected value the label "D", we have:

         D = (1/6)*3 + (1/6)*4 + (1/6)*5 + (1/2)*(D + 6)
         D = (12/6) + (D/2) + 3
 D - (D/2) = 2 + 3
       D/2 = 5
         D = 10

Of Course, I forgot about recursive functions. That is what happens when I do sleepy math.

Following this:

 D = 1/8*(3+4+5+6+7) + 3/8 * (D+8)
5/8 D = 49/8
D = 49/5 = 9.8

D = 1/10*(3+4+5+6+7+8+9) + 3/10 * (D+10)
7 / 10 D = 72/10
D = 72/7 = 10.2857143

D = 1/12*(3+4+5+6+7+8+9+10+11) + 1 / 4 * (D+12)
3 / 4 D = 99/12
D = 396/36 = 11

I'll update the first post.

General Case: DX, X != 0
D = 1/X * (sum from 3 to X-1) + 3/X * (D + X)
XD = (X)(X-1)/2 - 1 - 2 + 3D + 3X
2(X-3) D = (X^2+5X-6) = (X+6)(X-1)
D = (X+6)(X-1)/(2X-6)

So, for a d20 weapon, your looking at 14.5294118. For a d100 weapon, your looking at 54.0927835.

[Hal]

I was told there would be no maths in this game.

Also, I don't mind high level combos like this. It's a fun part of the game to find neat combos like that, and it's frustrating every time they get errata'd away. Although let's not confuse "fun" with "ridiculously broken."

[tcrudisi]

I was told there would be no maths in this game.

Also, I don't mind high level combos like this. It's a fun part of the game to find neat combos like that, and it's frustrating every time they get errata'd away. Although let's not confuse "fun" with "ridiculously broken."

This is far from "ridiculously broken" at level 29. It's a little bit more powerful than the average for the level, but by no means broken. Broken would be NI combos (which are attainable at level 30). This is nowhere close to NI.