TwoSix
Dirty, realism-hating munchkin powergamer
The OOV paladin's DPR is crazy high...what assumptions are baked into those numbers? I'm assuming those are leveraging divine smite, especially on crits, right?
D&D 5E GWF vs. TWF Fighting styles
He actually gave you a weighted arithmetic mean formula, not the "standard" average.
(1+2+3+4+5+6+1+2+3+4+5+6+3+4+5+6)/16
is just
(6*3.5 + 6*35 + 1*3 + 1*4 + 1*5 + 1*6) / (6 + 6 + 1 + 1 + 1 + 1)
In other words, the exact weighted average formula.
If you want a "standard" average, there are actually 36 events to consider, not 16. You can abstractly think of the damage as a 1d6 roll followed by another 1d6 roll, where 3s, 4s, 5s, and 6s in the first roll will do a second roll with 6-sided dice where the faces all the same number. Essentially, you have 6 * 6 rolls, or 36 rolls.
If you roll a 1, you get 1, 2, 3, 4, 5, or 6.
If you roll a 2, you get 1, 2, 3, 4, 5, or 6.
If you roll a 3, you get 3, 3, 3, 3, 3, or 3.
If you roll a 4,... well, etc.
So, if we want the probability of a 3, it's 8/32 or 22.2%, precisely what is expected. The formula you provided says something to the effect of "the probability of rolling a 3 is 3/16ths or 18.75%" which we know is false.
So the average is
(1+2+3+4+5+6 + 1+2+3+4+5+6 + 3+3+3+3+3+3 + 4+4+4+4+4+4 + 5+5+5+5+5+5 + 6+6+6+6+6+6) / (6 + 6 + 6 + 6 + 6 + 6)
or simplified
((1+2+3+4+5+6)/6 + (1+2+3+4+5+6)/6 + 3 + 4 + 5 + 6)/6
which is just algebra for Average(3.5,3.5,4,5,6) like was posted earlier.
This is all just simple math to show the logic behind Average(3.5,3.5,4,5,6). The difference is one formula assumes 16 events of equal probability and the other assumes 36.
Incidentally, I made a model in C++ that does 1 million random dice rolls (uniform distribution, reroll 1s and 2s once), and the results matched, ~5.5% for 1s and 2s, ~22.2% for 3s-6s, and a 4.167 average.
I stopped lurking and made an account for a math post. Dunno what that says about me.
TwoSix
Dirty, realism-hating munchkin powergamer
You're an awesome poster for helping to stem the tide of dyscalculia, of course.
Kryx
Explorer
Yes, it does.
See GWF Paladin Reroll 1s & 2s on Divine Smite? I was against it. I was wrong.
Without it applying on additional dice Defensive Fighting Style is much better for Paladin and Battle Master.
Just to pile on:
It is quite straightforward to write a program to literally roll a d6, then reroll on a 1-2. In Python, for example, you can do it very few lines
In any case, the original reasoning of replacing the rerolls with the average of 1d6 was completely correct.
Not that it matters really, but I am by trade a computational physicist who does this stuff for a living. And the most brilliant mathematician I've known was also utterly incapable of multiplying two numbers correctly... his thoughts were far more abstract.
It is quite straightforward to write a program to literally roll a d6, then reroll on a 1-2. In Python, for example, you can do it very few lines
Code:
In [1]: d = randint(1,7,1e6) # 7 here because Python is quirky and doesn't include the top number
In [2]: d2 = randint(1,7,1e6)
In [3]: d[where(d < 3)] = d2[where(d < 3)] # use the second roll where the first is one or two
In [4]: mean(d)
Out[4]: 4.1659709999999999In any case, the original reasoning of replacing the rerolls with the average of 1d6 was completely correct.
Not that it matters really, but I am by trade a computational physicist who does this stuff for a living. And the most brilliant mathematician I've known was also utterly incapable of multiplying two numbers correctly... his thoughts were far more abstract.
Kryx
Explorer
Paladin does really good damage. Especially OoV.
I'm not sure where to start, but I'll just rant. Lets take Polearm for example as that is more straightforward. Lets do level 20 to make it easy. I'll assume the RAW numbers
First their is straight weapon DPR which is calculated by chance to hit multiplied by damage - shouldn't be any issue there. A normal Paladin would do 22.7 weapon damage (crit, -5/+10, and IWD included) after a 40% chance to hit between 2 attacks. For provoke take the normal weapon damage and multiple it by the chance of a provoke. I assume that happens 40% of the time. That means on the average 5 round encounter an enemy enters your reach and you have a reaction 2 out of 5 rounds. The provoke is worth 4.5 DPR. A bonus attack is worth 9. I normally assuming an opportunity attack chance of 12.5% (slightly above once out of every 10 rounds), but since the provoke takes the reaction 40% of the time I multiple 12.5x.6 to get 7.5% of the time for 0.9 DPR (who cares). His smite is based on the total amount of smite damage divided by the amount of rounds in a day. I assume he doesn't use his level 5 spells for smite as they do not scale past 5d8. I also assume he uses 15% of his spells on things other than smite. With that and GWF he would do 17.8*43%*some crit stuff to get 8.9 DPR.
So the normal paladin does 46 DPR.
A OoV Paladin has hunter's mark so I assume that in. That is 3.53 attacks multiplied by the normal to hit and everything. To model the bonus action economy I assume the following situation: Round 1 hunters mark. round 2 bonus attack. Round 3 hunter's mark. Round 4 bonus attack, etc. That is a 50% chance of having hunter's mark up. Though honestly there could be a lot more done here. For example I still assume the bonus attack goes off on all rounds. I also ignore concentration which is basically likely to fail by level 6+ and surely to fail by 9+ if you are hit (average damage is enough for DC 20 at CR6, or DC 30 at CR 9+). I need to fix this up a bit if I can. For now I have 5.14 DPR from 3.53 attacks. It would be less.
In total that's 51 DPR for OoV Paladin (not including Oath of Enmity)
Now for Oath of enmity: I calculate all the same stuff with advantage. His total is 66.5 DPR. To get a total based on resources expended I have 3 uses of oath of enmity lasting for 5 rounds each. However you could kill the creature you've targeted so I assume 2.5 rounds (half the time 2, half 3). 2.5*3 = 7.5 out of a total of 25.7 rounds on average = 29% of the time on an average adventuring dat the paladin is using OoE.
So I then multiply (1-.29)*51 + .29*66.5 to get 56 DPR.
Hopefully that has been helpful. Please do let me know if you have any issues with the math or assumptions. I'd like to improve the sheet as much as possible.
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Prism
Explorer
Did any of the examples in your spreadsheet assume Great Weapon Fighting without using a polearm? Most people don't bother with polearms so I was wondering how different it was to sword and board or TWF. Also polearm use typically uses two feats whereas the other styles one or no feats.
Prism
Explorer
Ah yes, found it. Looking on the wrong sheet.
We play with magic items and to some degree don't use polearms since magic versions are pretty uncommon. They change things quite a bit at higher level but admittedly impossible to calculate for.
Looks like GWM is the factor that effects things the most. Then again one of my characters typically does slightly less damage when using it due to a magic weapon weighting the damage
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