Attila's math desk, now open for business!

[Archmage1]

Hey... Just noticed, but nice sig, Atilla.

[AttilaTheGeek]

Hey... Just noticed, but nice sig, Atilla.

[yougi]

About the rolling 2d20b1, I got an average of 14 (3.5 higher average on 1d20). However, it also almost doubles the odds of a critical hit. My idea is to create a weapon enchantment that would allow you to do it, and to price it accordingly. How could I calculate the average damage of such a weapon?

[Randomguy]

One day, I hope to be able to do this math.


There's an exotic weapon in the 3.5 MM4 called the Minotaur Greathammer. It deals 1d12 damage with a threat range of 19-20 and a crit multiplier of x4.

What's the Minotaur Greathammer's average damage? How does it compare to the average damage of the greatsword, greataxe and falchion?


If you have a friendly wizard to cast Greater Mighty Wallop on your minotaur greathammer each day, then how would the hammer's damage output compare to that of a heavy flail, if you also had the same wizard casting the spell on your flail?
Could I please get a comparison for caster levels 5, 8, 12 and 16 on the mighty wallop spell? Assume a medium sized wielder.

[Story]

What's your attack bonus and target AC? Critical hit damage depends strongly on what roll you need to confirm.

[TypoNinja]

What is the operation I need to preform to determine the probability of any one of two (or more) events happening.

For example, say I have two passive abilities that each have a 20% chance to activate in a combat round.

What about 3, or 4 abilities?

I know that I multiply the % together to get the odds of both happening, but I've no idea what operation or formula to apply in this case.

[Story]

If the events are independent and occur with probabilities p and q, then

The chance of both occuring is pq
The chance of neither occuring is (1-p)(1-q)
The chance of exactly one happening is p(1-q) + q(1-p).

If you want to know the probability of one or more events occuring with individual probabilities p1, p2, ...

Then it is 1 - (1-p1)(1-p2)(...)

[AttilaTheGeek]

One day, I hope to be able to do this math.

There's an exotic weapon in the 3.5 MM4 called the Minotaur Greathammer. It deals 1d12 damage with a threat range of 19-20 and a crit multiplier of x4.

What's the Minotaur Greathammer's average damage? How does it compare to the average damage of the greatsword, greataxe and falchion?

The set of rolls on 1d12 is {1,2,3...10,11,12}, which averages 6.5. In general, a roll of 1dX for any X will average 1/2*X + 1/2. A crit multiplier of x4 really means that x3 damage is added on a crit, because the first damage would have been done anyway. Like Story said, crit confirmation rolls depend on how likely you are to hit in the first place, which depends on your target AC and the attack bonus, B. (AC-B) is the number you need to roll on your d20 to hit, and (AC-B)/20 is the chance of rolling that number or below. For example, you have a 15/20 chance of rolling a 15 or below. 1 minus that is the chance of not rolling that number or below, but we add 1/20 bcause an equal attack hits, so the total chance of hitting is 1.05-(AC-B)/20. The extra damage added because of crits is the likelihood of a crit times the damage added. The chance of you getting a crit after you've already hit is the chance of threatening (2/20 for this 19-20 weapon) and confirming (times 1.05-(AC-B)/20), and the added damage is 3 times 6.5 makes 19.5. Then we multiply out to find that the damage added becaue of crits is 19.5*2/20*(1.05-(AC-B)/20.), or 2.0475-0.0975(AC-B). We add the base damage to that (6.5) to get total damage on hit of 8.5475-0.0975(AC-B) It may look strange, but the "reality check" gives it a pass- higher AC decreases damage, and because of the double negative, a higher attack bonus increases it.

A Greatsword hits for 2d6, averaging 7. Damage added because of crits is 7*2/20*(1.05-(AC-B)/20), which multiplies out to 0.735-0.035(AC-B). Added together, a Greatsword's average hit will do 7.7735-0.035(AC-B) damage.

A Greataxe's 1d12 averages to 6.5, and the damage added from crits is 6.5*2*1/20*(1.05-(AC-B)/20). That multiplies to 0.6825-0.0325(AC-B). Added together, a Greateaxe hits for 7.1825-0.0325(AC-B).

And, last but not least, the falchion. 1d4 averages 2.5, so 2d4 averages 5. It has a 3/20 chance to threaten a crit, adds 1 extra instance of 5 damage on a crit, and has the same chance to confirm a crit as every other weapon. 5*1*3/20*(1.05-(AC-B)/20) makes 0.7875-0.0375(AC-B). Added together, a falchon's damage is 5.7875-0.0375(AC-B).

If you have a friendly wizard to cast Greater Mighty Wallop on your minotaur greathammer each day, then how would the hammer's damage output compare to that of a heavy flail, if you also had the same wizard casting the spell on your flail?
Could I please get a comparison for caster levels 5, 8, 12 and 16 on the mighty wallop spell? Assume a medium sized wielder.

Sure thing.Though I assume you meant CLs 4, 8, 12, and 16, because it scales every 4 levels.

{table=head]CL|Greathammer damage|Greathammer average damage|Flail damage|Flail average damage
None|1d12|8.5475-0.0975(AC-B)|1d10|6.0775-0.0275(AC-B)
4|3d6|12.705-0.105(AC-B)|2d8|9.945-0.045(AC-B)
8|4d6|16.94-0.14(AC-B)|3d8|14.9175-0.0675(AC-B)
12|6d6|25.41-0.21(AC-B)|4d8|19.89-0.09(AC-B)
16|8d6|33.88-0.28(AC-B)|6d8|29.835-0.135(AC-B)[/table]

The Greathammer wins. If you want to talk about math or ask for help, feel free to PM me

[AttilaTheGeek]

What is the operation I need to preform to determine the probability of any one of two (or more) events happening.

For example, say I have two passive abilities that each have a 20% chance to activate in a combat round.

What about 3, or 4 abilities?

I know that I multiply the % together to get the odds of both happening, but I've no idea what operation or formula to apply in this case.

In addition to what Story said, if you have X abilities, and each of them has the same chance P of activating on any given weapon hit, then the chance of any one of them activating is 1-((1-P)^X). If you have four abilities that each have a 20% chance, the chance of getting at least one of them is 1-((0.8)^4), or 59.04%. For three abilities it's 48.8%, and for two it's 36%.

[Stouts]

About the rolling 2d20b1, I got an average of 14 (3.5 higher average on 1d20). However, it also almost doubles the odds of a critical hit. My idea is to create a weapon enchantment that would allow you to do it, and to price it accordingly. How could I calculate the average damage of such a weapon?

I haven't run numbers on damage output, but a quick look at the distribution (click 'graph' for ease of viewing) shows that this would probably be better than +3 Enhancement for attack and better than Keen for damage. I'd ballpark it at +4.

[Pandoras Folly]

Ok easy one. My human character was just made a Large size character by dm fiat. What changes? I know I have a 10 foot reach, but I'm not real sure after that.

[Stouts]

Ok easy one. My human character was just made a Large size character by dm fiat. What changes? I know I have a 10 foot reach, but I'm not real sure after that.

The DM obviously is going to control the particulars, but it's likely that you just get the listed effects of enlarge person. Even if the DM doesn't give you the stat changes, though, large always carries a -1 hit to Attack and AC. You can also use that link as a reference for appropriately sized weapon damage.

[Twilightwyrm]

Alright. I will take on this challenge. I want to know what level I should run the test for (5, 6, or 7), what feats you'd have by then, what extra fighter feats you'd have, what rogue talents you would take, and what archetypes you'd take.

(Sorry it took so long to respond)
Sure, I can provide this. The level test should be for 12th level (assume Gunslinger 7/Fighter 5 or Gunslinger 5/Rogue 7).

By 12th level for the Gunslinger/Fighter Hybrid, I'm expecting the following: Archetypes: Musket Master (Gunslinger) and Weapon Master (Fighter)
Feats: Skill Focus (Stealth)*, Quick Draw, Gunsmithing**, Rapid Reload (Musket)***, Deadly Aim, Point Blank Shot**, Signature Deed (Deadeye), Improved Initiative, Rapid Shot****, Signature Deed (Targeting), Improved Critical****, Hammer the Gap, Snap Shot.

By 12th level for the Gunslinger/Fighter Hybrid, I'm expecting the following:
Archetypes: Musket Master (Gunslinger) and Sniper (Rogue)
Talents: Snap Shot, Bleeding Attack, Major Magic (Vanish)
Feats: Musket Master (Gunslinger) and Weapon Master (Fighter)
Feats: Skill Focus (Stealth)*, Quick Draw, Gunsmithing**, Rapid Reload (Musket)***, Deadly Aim, Point Blank Shot**, Rapid Shot, Signature Deed (Deadeye), Precise Shot, Improved Initiative
Additional Items: Sniper's Goggles (+2 damage per SA die (or +8 damage on SAs))

Note: Assume any other equipment, special abilities, or feats I've forgotten do not effect damage per round, and all other factors being equal.
Feel free to request additional information you might find pertinent.
*Half-Elf Bonus Feat
*Gunslinger Bonus Feat
***Musket Master Bonus Feat
****Fighter Bonus Feat

[AttilaTheGeek]

I haven't run numbers on damage output, but a quick look at the distribution (click 'graph' for ease of viewing) shows that this would probably be better than +3 Enhancement for attack and better than Keen for damage. I'd ballpark it at +4.

(emphasis mine)

Actually, I had a hunch that wasn't true, so I decided to crunch some numbers.

Spoiler

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If you take C, the chance to Crit on 1d20, then the chance to not crit on 1d20 is (1-C). The chance to not crit on either of 2d20 is (1-C)^2, so the chance to crit on either of them is 1-(1-C)^2. However, the chance to confirm the critical is the chance to hit naturally, so let's call that H for Hit chance. The chance to confirm it with no enhancement or with Keen is H, but the probability of confirming a crit with advantage is 1-(1-H)^2.

{table=head]Crit range|Crit chance|With Keen|With Advantage|H must be
20-20|0.05*H|0.1*H|0.0975*[1-(1-H)^2]|97.4%
19-20|0.1*H|0.2*H|0.0199*[1-(1-H)^2]|94.7%
18-20|0.015*H|0.3*H|0.2775*[1-(1-H)^2]|91.8%[/table]

Then, I found out what the hit chance had to be for Advantage to be better. If every weapon swing hits, then every critical threat confirms, and Keen is better because Keen weapons have a larger chance to threaten. However, an Advantage weapon is probably more likely to hit, and so more likely to confirm the crit. I took the third column of each row in the table above and subtracted the second column to find the additional chance to crit that an Advantage weapon would have over a Keen weapon. Then, I graphed that as a function of H from 0 to 1. Wherever the function was positive, Advantage was better, and the value of that function was the extra damage added by Advantage over Keen in terms of the weapon's base damage. I then calculated the zero of the function to find out what the hit chance had to be for Advantage to be worse than Keen. That's how I generated the last column, what H must be for Keen to hit harder.

Conclusion: My hunch was wrong. Advantage is almost always better than Keen for damage. Assuming your initial attack roll hits, Keen only hits harder than Advantage when you have an 18-20 crit range and you can confirm the crit with a 3 on a d20 roll. A +4 price seems good.

[CockroachTeaParty]

What I'm taking away from this thread so far is that crit-fishing builds are probably a bad idea.

[AttilaTheGeek]

Oh yeah. It's common knowledge among optimizers that the 1d6 energy enhancements are solid but not great, and for an x4 crit weapon or an 18-20/x2 weapon every single hit needs to do over 20 damage for Keen to even catch up to that d6.

[Sugashane]

I love this idea.

Ok here is one that is confusing me, and please show the breakdown for me.

My large LVL 12 fighter with 30 STR (+10) using a large greatsword (3d6 +15)

Feats include:
Power attack (+12)
Shock trooper's heedless charge (why above can go to +12)
Leap attack modifier
Battle Jump (Unapproachable east)
________________________________

What would my average attack be if charging?
And would having a valorous weapon be more damaging or a magic item that allowed for pounce?

From what I understand the power attack is double with two hands (26 (12+2x2)) with the extra 2 being from charging which is "tripled" by leap attack (D&D math making it now a x4 multiplier) and again doubled by Battle Jump (so really x5). So would it be 3d6+ 70 (Bonus)+15 (STR) for an average of 95.5 damage?

Thank you, the errata and conflicting posts online have been frustrating to say the least.

[yougi]

I have a good one: what is the average of 5d6, keep the best 3? And while we're at it, when we roll 4d6 (Best 3) 8 times and keep the highest 6 scores, what would be the average of that? How do these methods compare, and what kind of point buy would they be equivalent to (assuming D&D Point buy, not PF).

[AttilaTheGeek]

I love this idea.

Ok here is one that is confusing me, and please show the breakdown for me.

I think that's beyond the scope of this thread. Because of the rules errata and conflicting rules, like you mentioned, that would go better in the simple Q&A thread, or perhaps in its own thread. I can tell you that the 3d6 averages 10.5, so you can add that to whatever constant you come out with to find out how hard you'll hit.

[Umbranar]

I have a question (D&D 3.5 btw):

My character has a two weapons to choose from:

+1 keen frost bastard sword - +16/+11 to hit 1d10+11 damage +1d6 frost
+2 Greatsword - +17/+12 to hit 2d6+12 damage

Knowing my avarage opponent has an AC of 24, which weapon is better and how great a penalty should I take with power attack to have the best effency (best damage per round)?

extra info for funz if you feel like it: im a dragon disciple and I have a bite attack which I add to full round attacks:

Bite - +10 to hit 1d6+3 damage

[CockroachTeaParty]

I was curious about something:

What over the long-term is statistically the superior one-handed exotic weapon in Pathfinder?

Bastard Sword: 1d10, 19-20/x2

Katana: 1d8, 18-20/x2

Falcata: 1d8, 19-20/x3

[yougi]

I have a question (D&D 3.5 btw):

My character has a two weapons to choose from:

+1 keen frost bastard sword - +16/+11 to hit 1d10+11 damage +1d6 frost
+2 Greatsword - +17/+12 to hit 2d6+12 damage

Knowing my avarage opponent has an AC of 24, which weapon is better and how great a penalty should I take with power attack to have the best effency (best damage per round)?

extra info for funz if you feel like it: im a dragon disciple and I have a bite attack which I add to full round attacks:

Bite - +10 to hit 1d6+3 damage

I can answer that.

As for which one is the best, well against an AC of 24:

Bastard Sword has average damage of 20 (5.5+11+3.5), and Greatsword has 19 (7+12) on a non-crit hit.

The bastard will deal an average of (13/20*20)+(4/20*13/20*16.5)+(8/20*20)+(4/20*8/20*16.5) = 24.465 points of damage per round.

The Greatsword will deal an average of (14/20*19)+(2/20*14/20*19)+(9/20*19)+(2/20*9/20*19)= 24.035 points of damage per round.

The difference is very small, but Bastard sword is better.

POWER ATTACK
Against a target with an AC of 24, considering you use the Bastard Sword from before, you're better off not power attacking on a full attack, and power attacking at -2 on a single attack.

Math:

Spoiler

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FULL ATTACK
No Power attack (including bite): (13/20*20)+(4/20*13/20*16.5)+(8/20*20)+(4/20*8/20*16.5)+(7/20*6.5)+(1/20*7/20*6.5)= 26.853

Power Attack at -1: (12/20*22)+(4/20*12/20*18.5)+(7/20*22)+(4/20*7/20*18.5)+(6/20*7.5)+(1/20*6/20*7.5)= 26.775

So on a full attack, power attacking lowers you damage per round.

SINGLE ATTACK
No Power Attack: (13/20*20)+(4/20*13/20*16.5)= 15.145 pts of damage

Power Attack at -1: (12/20*22)+(4/20*12/20*18.5)= 15.42

Power Attack at -2: (11/20*24)+(4/20*11/20*20.5)= 15.455

Power Attack at -3: (10/20*26)+(4/20*10/20*22.5)= 15.25

So at -3, it starts going down again.

[AttilaTheGeek]

I was curious about something:

What over the long-term is statistically the superior one-handed exotic weapon in Pathfinder?

Bastard Sword: 1d10, 19-20/x2

Katana: 1d8, 18-20/x2

Falcata: 1d8, 19-20/x3

Math:

Spoiler

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Even if you always cofirm crits, the Falcata hits for 1d8 (4.5) times (1+ 2/20 * 2) = 5.4. The 1+ comes from the fact that it always does normal damage when it hits, the 2/20 is the crit chance, and the 2 is the extra weapon damage added by a crit.

The Katana is just worse than the Falcata because the katana's critical threat range adds 1x weapon damage 3/20 of the time (for a total of 3/20 extra damage before critical hit confirmation), whereas the Falcata's crit range adds 2x extra weapon damage 2/20 of the time for a total of 4/20 extra damage.

Even if the Bastard Sword never crit, its 1d10 damage still averages 5.5, which is above the Falcata's 5.4 even if it confirms all crits. Besides, rolling d10s is fun.

Conclusion: Bastard Sword, by a fairly large margin.

[TuggyNE]

Conclusion: Bastard Sword, by a fairly large margin.

This is only true if you have no extra damage that can be multiplied on a crit; it doesn't take all that much extra to make the falcata superior.

[AttilaTheGeek]

(Sorry it took so long to respond)
Sure, I can provide this. The level test should be for 12th level (assume Gunslinger 7/Fighter 5 or Gunslinger 5/Rogue 7).

I will, uh...get to it when I have a free two hours?

My gut says the Fighter, but only because getting off 75% sneak attacks with a ranged weapon is nearly impossible unless you have a mechanic that can make people flat-footed, like Dazzling Display and Shatter Defenses. Do you want me to model it with something more realistic, or are you confident in the 75% sneak attacks? If you are, it's going to be a close race.

[AttilaTheGeek]

This is only true if you have no extra damage that can be multiplied on a crit; it doesn't take all that much extra to make the falcata superior.

Oh. Right. I forgot about that. And he did say "long-term". I'm going to make the assumption that our character needs an 8 on a d20 roll to hit because it's reasonable across most levels and lets us get an answer just in terms of how much damage you need for the falcata to be better.

Spoiler

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Our damage is in terms of the Bonus damage, B. The Bastard Sword does 5.5+B damage on a normal hit with an additional 5.5+B on a crit, which happens (2/20) * (12/20), or 24/400 of the time. Adding those together, we get 5.5+B + 24/400 * 5.5+B, or (5.5+B)*424/400. That multiplies to 5.83+1.06B damage on a hit.

The Falcata hits for 4.5+B, and an extra 9+2B when a crit is threatened and confirmed on (3/20)*(12/20) of hits, or 36/400 of all hits. 4.5+B+ (9+2B)*36/400 multiplies to 4.5+B+0.81+0.18B. Adding that all together, we get 5.31+1.18B damage on hit.

Then, to find out what damage we need to make the falcata more powerful, we put them into an inequality and solve.
5.31+1.18B > 5.83+1.06B
0.12B > 0.52
B>4.3

You were right. As long as you add 5 or more damage to every attack, the Falcata is better. So, in the long run, the Falcata is the winner, not the Bastard Sword.

[Umbranar]

I can answer that.

As for which one is the best, well against an AC of 24:

Bastard Sword has average damage of 20 (5.5+11+3.5), and Greatsword has 19 (7+12) on a non-crit hit.

The bastard will deal an average of (13/20*20)+(4/20*13/20*16.5)+(8/20*20)+(4/20*8/20*16.5) = 24.465 points of damage per round.

The Greatsword will deal an average of (14/20*19)+(2/20*14/20*19)+(9/20*19)+(2/20*9/20*19)= 24.035 points of damage per round.

The difference is very small, but Bastard sword is better.

POWER ATTACK
Against a target with an AC of 24, considering you use the Bastard Sword from before, you're better off not power attacking on a full attack, and power attacking at -2 on a single attack.

Math:

Spoiler

Show

FULL ATTACK
No Power attack (including bite): (13/20*20)+(4/20*13/20*16.5)+(8/20*20)+(4/20*8/20*16.5)+(7/20*6.5)+(1/20*7/20*6.5)= 26.853

Power Attack at -1: (12/20*22)+(4/20*12/20*18.5)+(7/20*22)+(4/20*7/20*18.5)+(6/20*7.5)+(1/20*6/20*7.5)= 26.775

So on a full attack, power attacking lowers you damage per round.

SINGLE ATTACK
No Power Attack: (13/20*20)+(4/20*13/20*16.5)= 15.145 pts of damage

Power Attack at -1: (12/20*22)+(4/20*12/20*18.5)= 15.42

Power Attack at -2: (11/20*24)+(4/20*11/20*20.5)= 15.455

Power Attack at -3: (10/20*26)+(4/20*10/20*22.5)= 15.25

So at -3, it starts going down again.

I thank you very much, now maybe for an extra: We have a bard who manages to boost our attack by +6 to hit +7 damage (inspirational boost, masterwork instrument, words of creation and badge of valor). How does this change the math. Also I have boots of Speed or I am regularly buffed with haste. How does that change things

[Squirrel_Dude]

A pretty simple one, i think, but I don't want to try and do it.


What's the chance of a Level 20 monk in Pathfinder hitting a Pit Fiend (AC 38) 4 times using flurry.
- Monk has 28 strength
- +5 Amulet of Might Fists
What is the most likely number of times that the monk will hit the Pit Fiend?

What's the chance of a Level 20 Fighter in Pathfidner hitting a Pit Fiend 4 times using his full attack.
- Fighter has 28 Strength
- Fighter has +5 bonus to attacking with this weapon (weapon focus + weapon training 1)
What is the most likely number of times that the fighter will hit the Pit Fiend?

[CockroachTeaParty]

As long as you add 5 or more damage to every attack, the Falcata is better. So, in the long run, the Falcata is the winner, not the Bastard Sword.

Interesting. Thanks for the maths!

Now, which one to pick for my character... he's a STR-based, shocking grasp-channeling magus, by the way, so I'm not sure if that changes anything.

[AttilaTheGeek]

Interesting. Thanks for the maths!

Anytime.

Now, which one to pick for my character... he's a STR-based, shocking grasp-channeling magus, by the way, so I'm not sure if that changes anything.

In that case, I'd probably go with the Katana, because 50% more crits on your ridiculous, ridiculous Shocking Grasp is what you want. The Falcata would still be better, except for this:

This attack uses the weapon's critical range (20, 19–20, or 18–20 and modified by the keen weapon property or similar effects), but the spell effect only deals ×2 damage on a successful critical hit, while the weapon damage uses its own critical modifier.

With Shocking Grasps reaching into the bajillions, the weapon damage is pretty much negligible in comparison, so the 19-20/x3 crit range really acts like 19-20/x2.