I believe my exact words were that an 18-20 crit range gives me 5% more damage than a 19-20 crit range.
This is a statement I see frequently repeated(both on the forums and IRL). Its wrong. If you are currently hitting on an 11-20, a +1 to hit will cause you to do 10 percent more damage, because you will be hitting 10 percent more often. The fact that you are using a 20 sided die is irrelevant. You could roll a 12 sided die or a 100 sided die and a +1 to hit would still give you 10 percent more damage.
How exactly does hitting 1/20th more often translate to hitting 10% more often?
1/20 is .05
1/19 is .052
Unless I'm missing something by your logic my 5% extra was a generous assumption.
How exactly does going from 10/20 to 11/20 equal to a 5 percent increase?
7% and 5% are very close when someone's making estimates.
Pretty sure most people aren't sitting there with calculator in hand when they do these calculations.
I see what he's saying. Let's say you have a +5 to hit and you need to hit an AC 16. It'd require you to roll an 11-20, essentially a 50% chance to hit that AC. If you add a +1 to your to-hit, that becomes a 10-20, or a 55% chance to hit. However, looking at the change from 11-20 to 10-20, it's technically a 10% change because that +1 is 10% of the ten numbers you would use to hit someone.
F+~%ing statistics man.
Well .052 is greater than 5 percent and that assumes you are fighting monsters with no AC.1/20 is .05
1/19 is .052
Unless I'm missing something by your logic my 5% extra was a generous assumption.
We'll assume the monster has a 20 AC.
The weapon with a 19-20 crit range will crit 2/20, or 1/10 of the time.
Say a Level 10 Fighter, so he'll have 10 BaB and a +5 Str mod, meaning he has to roll a 5 to hit the monster. So he crits on 2/15 possible rolls.
If he's using an 18-20 weapon he crits on 3/15 of those rolls.
2/15= .13
3/15= .2
Unless I've messed up, that's still 7%. Not 10%. That 7% is a lot closer to my 5 than your 10 I'll tell you that.
The to hit/AC thing is probably spot on though.
If you're only hitting on a 20, then +1 to hit means 100% more damage, crits notwithstanding.
Of course, it's not always 10%. In fact, the more you add to your 'to-hit' ability, the less of an increase in overall to-hit there is. Similar example, with a +6 to your to-hit against an AC 15, you need a 9-20 to hit (60% chance to hit). Add a +1 to that and you get a 8.3% increase to your to-hit value, from 9-20 to 8-20
Yeah hes not arguing any exact amount he is just saying the generic "+1 to hit is 5% more damage!" statement is well wrong. In which case he is right. The value of to hit is so wide that any generic statement about it will most of the time be wrong as it will change from one enemy to the next.
The problem is that that's not the only case there is, and likely not the most, well, likely case at that. Hitting on an 11-20 seems pretty low for anyone swingin' in melee combat (at least for a first hit).
I'm not saying you're wrong (you're not, exactly) and I'm not saying that blanket statement is right (it's not) but I don't think you're exactly 100% right either.
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I've actually never heard that +1 to hit is 5% to damage, but 5% to hit.
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It's an increase of five percentage points.
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Johnlocke90 is correct.
The increase of damage when your hit will depend on what you need to roll in order to make a hit.
Also note that if you were already hitting on a roll of 2 that +1 to hit does not increase your damage (for that attack).
Well .052 is greater than 5 percent and that assumes you are fighting monsters with no AC.1/20 is .05
1/19 is .052
Unless I'm missing something by your logic my 5% extra was a generous assumption.
We'll assume the monster has a 20 AC.
The weapon with a 19-20 crit range will crit 2/20, or 1/10 of the time.
Say a Level 10 Fighter, so he'll have 10 BaB and a +5 Str mod, meaning he has to roll a 5 to hit the monster. So he crits on 2/15 possible rolls.
If he's using an 18-20 weapon he crits on 3/15 of those rolls.
2/15= .13
3/15= .2Unless I've messed up, that's still 7%. Not 10%. That 7% is a lot closer to my 5 than your 10 I'll tell you that.
The to hit/AC thing is probably spot on though.
Actually you've messed that up completely
Lets say this fighter does 10 points of damage on a hit, so 20 on a crit (not trying to be realistic, just making the math easier)
in this case the fighter has:
20% chance of not hitting the monster (1-4)
70% chance of hitting 5-18
10% chance of a threat (19-20)
the threat can be broken down further into confirming and not confirming.
20% chance of not confirming
80% chance of confirming
So after taking crits into account we have:
20% chance of 0 damage (miss)
72% chance of 10 damage (hit or not confirmed crit)
8% chance of 20 damage (crit)
this gives us an average of 8.8 damage.
Going through the same steps with an 18-20 weapon will give us
20% chance of 0 damage (miss)
68% chance of 10 damage (hit or not confirmed crit)
12% chance of 20 damage (crit)
giving us an average of 9.2 damage
the increase of damage is close to 4.545%
Also going from 13% to 20% is not a 7% increase, it's more like a 54% increase.
That's not going from a 13 to a 20% increase, that is increasing the chance of a critical.
I.E. your 19-20 guy has a 13% chance to crit, whereas your 18-20 guy has a 20% chance to crit. Which is a flat 7% increase, even though relatively it is a 54% increase.
And again, 4.545% is a lot closer to my stated estimate of 5% than his stated value of 10%.
This is a statement I see frequently repeated(both on the forums and IRL). Its wrong.
Yes, it is "wrong".
It is also surprisingly accurate as a rule-of-thumb!
Make yourself a damage-per-round spreadsheet, make sure to parameterize it so that you can easily alter your assumptions (weapon enhancement, base attack, other attack boni, damage boni, imp crit feat etc etc).
I consulted one of my own DPR spreadsheets to refresh my memory for this post, and: Using the (unhasted) stats of my current party's fighter @ lvl 8, a +1 to hit bonus results in:
+0% damage vs AC 14
+2,2% dam vs AC 16
+5,7% dam vs AC 21
+6,1% dam vs AC 22
+8,9% dam vs AC 26
+10,7% dam vs AC 28
A quick look at CR 8 monsters in the Bestiary gives AC 22 for a young copper dragon, as the highest AC I would expect to meet at that level. NPC fighter might get higher, but the average combat monster encountered @ lvl 8 will have a lower AC than 22.
So if you are playing a "standard" scenario, the +1 = 5% heuristic will IME usually be an overestimate.
But this is all very sensitive to your assumptions!
Also... If your party's main melee'er gains a 100% damage boost from a +1 to-hit buff... which, as pointed out earlier, is theoretically possible... RUN! If you can; chances are you're already dead or captured.
While AC does not advance as rapidly as the AB of a combat character's first attack, it advances faster than the AB of the last iterative atk. Thus "+1 = 5%" holds relatively well for full attacks across the level spectrum... vs. highish AC anyway
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Average damage per hit (DPH): sum your max and min damage and divide by 2.
Probability of Hitting (PH) for an attack roll: (21-AC+Attack Bonus)*5%
Note: Never less than 5%.
CR = Critical Range (CR), in % (ie 19-20 is 10%)
CM = Critical Multiplier (CM), subtracting 1 (ie x2 means CM equals 1)
Probability of Critting for an attack roll (PC): CR * PH
Note: Assuming you always hit if you roll within your critical range.
Average damage per attack roll: (PH * DPH) + (PC * CM * DPH) =
= (PH * DPH) + (CR * PH * CM * DPH) =
= DPH * PH * (1 + CR * CM)
So if you increase your PH by 5%, your average damage increase by:
DPH * (PH + 5%) * (1 + CR * CM) - DPH * PH * (1 + CR * CM) =
= DPH * (1+CR*CM) * (PH + 5% -PH) =
= DPH * (1+CR*CM) * 5%
Assuming CR = 10% (19-20) and CM = 1 (x2):
Damage increase = 5% * DPH * (1 + 10% * 1) = 5% * DPH * 1,1 =
5,5% * DPH
Damage increase (%) = ((DPH * (PH+5%) * (1+CR*CM)) / (DPH * PH * (1+CR*CM))) -1 =
((PH + 5%) / PH) -1 = 5% / PH
So... a 7% or a 10% more damage does not make sense unless you define what is "damage", because:
The average damage per attack roll increases by 5,5% of your average damage per hit if you increase your probability to hit by 5%.
The average damage per attack roll increases by a percentage equal to 5% divided by your original probability to hit if you increase your probability to hit by 5%.
I think that maths are ok, but will be glad to read corrections on them
EDIT: Some wrong maths :P
The correct way to put it is that +1 to hit adds 5% of your average damage on a hit to your DPR.
For example, if you deal 100 damage with an average swing, getting +1 to hit will add +5 to your DPR.
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The correct way to put it is that +1 to hit adds 5% of your average damage on a hit to your DPR.
For example, if you deal 100 damage with an average swing, getting +1 to hit will add +5 to your DPR.
The entire point of this thread is to point out that that statement is incorrect.
I will prove by showing you the lowest increase a +1 to hit can give.
Lets say you can hit some one with a natural 3. That's 90% of your swings.
A +1 to hit means that you hit on a natural 2 now, 95%.
.95/.9 = 1.0555...
which is a 5.555...% increase in to hit, or average damage since average damage = chance to hit * average damage per swing.
By the exact wording of the title, it's wrong in all sense of the semantics and even the context.
+1 to hit does not mean you will do ANY more damage. Ever.
This might apply if you did a static amount of damage every round, but since you roll dice for damage, the logic is unsound.
Here's an example:
You attack with a 15, the Monster's AC is 15. You hit. You roll a 1 for damage. You do 1 damage.
You attack with a 16 (15 + your bonus of 1), the Monster's AC is 15. You hit. You roll a 1 for damage. You still only do 1 damage.
I assume you mean "A +1 to hit means I will hit 5% more often". This is roughly correct (.0526% chance, assuming natural 1 is always a miss).
+1 to hit does not mean you will do ANY more damage. Ever.
Any attack you would have missed by 1, the +1 will make you hit, and when you hit you do more damage than when you don't hit.
On average, you will do more damage per round.
I assume you mean "A +1 to hit means I will hit 5% more often". This is roughly correct (.0526% chance, assuming natural 1 is always a miss).
That looks wrong in multiple ways.
The question for all of these examples is, 5% of what?
If a +1 to hit increases my chance of hitting from 5% to 10%, that is an increase of +5% in one sense. In another sense, it is a +100% chance - it increases my chance of hitting to twice what it was before.
In terms of average damage per round, the '+100%' interpretation (ignoring crits) is the one to use.
The correct way to put it is that +1 to hit adds 5% of your average damage on a hit to your DPR.
For example, if you deal 100 damage with an average swing, getting +1 to hit will add +5 to your DPR.
The entire point of this thread is to point out that that statement is incorrect.
I will prove by showing you the lowest increase a +1 to hit can give.
Lets say you can hit some one with a natural 3. That's 90% of your swings.
A +1 to hit means that you hit on a natural 2 now, 95%.
.95/.9 = 1.0555...
which is a 5.555...% increase in to hit, or average damage since average damage = chance to hit * average damage per swing.
You're saying the same thing:
Let's take your example: 90% chance of hitting. Let's take a damage roll of 2d6+3. That's an average damage of 10 hp when he hits. Per round the expected damage is
90% * 10 = 9 hp
A +1 increase means that the expected damage is
95% * 10 = 9.5 hp
Increase in % = (9.5 / 9) -1 = 5.55 %
That's what you're saying
What he says is:
"adds 5% of your average damage on a hit" (not per round)
Your average damage on a hit is 10
5% of 10 is 0.5
"to the DPR"
The DPR is 9 so:
9 + 0.5 = 9.5
There are two positions on this issue: The OP's position, and the wrong one. It's math, people -- not philosophy.
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The correct way to put it is that +1 to hit adds 5% of your average damage on a hit to your DPR.
For example, if you deal 100 damage with an average swing, getting +1 to hit will add +5 to your DPR.
The entire point of this thread is to point out that that statement is incorrect.
I will prove by showing you the lowest increase a +1 to hit can give.
Lets say you can hit some one with a natural 3. That's 90% of your swings.
A +1 to hit means that you hit on a natural 2 now, 95%.
.95/.9 = 1.0555...
which is a 5.555...% increase in to hit, or average damage since average damage = chance to hit * average damage per swing.
You're saying the same thing:
Let's take your example: 90% chance of hitting. Let's take a damage roll of 2d6+3. That's an average damage of 10 hp when he hits. Per round the expected damage is
90% * 10 = 9 hp
A +1 increase means that the expected damage is
95% * 10 = 9.5 hp
Increase in % = (9.5 / 9) -1 = 5.55 %
That's what you're sayingWhat he says is:
"adds 5% of your average damage on a hit" (not per round)
Your average damage on a hit is 10
5% of 10 is 0.5
"to the DPR"
The DPR is 9 so:9 + 0.5 = 9.5
ah silly me,
I misread mplindustries' post as "100 damage average" rather than "100 damage per swing"
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This thread hurts my soul.
It is math as well as unimportant
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LOL. Math are hard.
It has been my experience that the vast majority of highly educated people have major difficulties understanding the most basic principles of statistics and probability.
This is in large part because there is a lack of understanding of the various frames of reference you can take to examine something statistically.
For example... Let's say you are playing Pathfinder and you currently are in a situation where your character hits an opponent by rolling an 11 or higher on a d20.
If you change that so that you are hitting on a 10 or better, what have you actually done in terms of increasing your frequency of hitting? Have you increased it by 5%?
Well, let's say you attack 200 times, 1/2 before and 1/2 after the adjustment. Statistically before the adjustment you would have hit 50 times. After the adjustment you would hit 55 times, so your frequency of hitting will have increased from 50% to 55%. So in ABSOLUTE TERMS you have increased your odds to hit by 5%.
But, you were hitting 50 times, now you are hitting 55 times. That's a TEN PERCENT increase in your frequency of hitting. So while your chance of hitting has increased 5%, your FREQUENCY of hitting has increased 10%.
What this means in terms of damage is complicated by things like frequency of critical hits and stuff.
This is exactly why a +2 on your attack is much more meaningful when you are having trouble hitting than when you almost always hit anyway.
If you are hitting 10% of the time, a +2 to your attack will double your frequency, which will roughly double your damage. If you are hitting 85% of your time a +2 will increase your frequency of hitting by about 11% and increase your damage by roughly the same percentage.
NOTE: This is why, if you have the choice to buff one of two melee combatants, it is frequently better to buff the one who is hitting LESS, not the one who is hitting MORE.
This is too much damn math.I was a math teacher for 2 years, and have a BS in Mathematics, and I cannot agree with this statement enough.
Nah... until you break out the partial differential equations, it's not nearly enough math.
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OK - here's a simpler question.
You have two weapons, both of which do the same base damage.
One of them threatens a critical on a 19-20, for 2x damage.
The other only threatens on a 20, but does 3x damage then.
Which increases your average DPR more (and by how much)?
How, if at all, is this affected if you would only hit on a natural 20?
OK - here's a simpler question.
You have two weapons, both of which do the same base damage.
One of them threatens a critical on a 19-20, for 2x damage.
The other only threatens on a 20, but does 3x damage then.Which increases your average DPR more (and by how much)?
** spoiler omitted **
So this would be a function where
D = damageP = probability to hit.
P(c) = probability to crit.
A(d) = Average damage
Weapon 1: A(d) = P*D + P(c)*2*D
Weapon 2: A(d)' = P*D + P(c)'*3*D
So A(d) - A(d)' = P(c)*2*D - P(c)'*3*D
Or A(d) - A(d)' = D*(P(c)*2 - P(c)'*3)
We know that P(c) = 2*P(c)' (To simplify I am assuming that all crits are confirmed and all damage is constant. In the full analysis both have the same probability to confirm a crit except for the edge condition I deal with separately, so the actual results are slightly different, but both weapons are impacted the same amount so the end result is the same)
So A(d) - A(d)' = D*(2*2*P(c)' - 3*P(c)')
Or A(d) - A(d)' = D*P(c)'
Which gives us A(d) = A(d)' + D*P(c)'
Since D*P(c)' has to be positive then A(d) > A(d)'
We can test this:
Let's plug in a probability of 50% to hit and 10 damage:
A(d) = .5*10 + .1*20 = 7
A(d)' = .5*10 + .05*30 = 6.5
Which shows that A(d)' is exactly D*P(c)' (.5) less in damage than A(d).
Now, boundary value conditions are always problematical. So this works for average rolls across a variety of armor classes, but only for situations that you can hit with less than a 20. That's because the probability of a crit is the same for both weapons when you need a 20 to hit.
When you look at the situation where you need a 20 to hit, then only a 20 can be confirmed to be a critical, because the rules specifically state that only a natural 20 always hits, no matter what your crit range is. So if a weapon crits on a 19, but a 19 is a miss, that's still a miss and so won't crit. And if a 19 is a miss, it also won't confirm a crit, so you still need two 20s to confirm the crit.
In that special case the second weapon does more damage.
^They're effectively the same.
Since I'm using both of these weapons on a character I'm building, we'll use the Earth Breaker and the Greatsword. They both deal 2d6 damage, but the former has x3 crit and the latter has 19-20/x2.
We'll assume a Fighter again, 5th level, +5 Str mod, assuming proficiency with both of course.
He hits on a 10 and above with both weapons.
With the Earthbreaker he crits on 1/10 of those swings.
With the Greatsword he crits on 2/10 of those swings.
20% of the time, the Greatsword will deal 4d6 damage (for simplicity's sake, we'll leave out Str mods and such), while 80% of the time he deals 2d6.
Out of 20 swings (assuming they miss half since they hit on a 10/20), he'll deal an average of 8 normal hits (16d6) and 2 critical hits (8d6) for an average of 84 damage.
The Earthbreaker, out of 20 swings, hits 9 regular hits (18d6) and 1 critical hit (6d6) for an average of 84 damage.
If they only hit on a 20, they'd both hit 1/20 hits, or 5% of the time, except the Earthbreaker would now deal 21 damage (6d6) while the Greatsword would only deal 14 (4d6).
Pretty sure I did all that right.
Edit: Dammit Adamantine.
Heh, actually I had an error in even my simplified probability equations.
I had:
Weapon 1: A(d) = P*D + P(c)*2*D
Weapon 2: A(d)' = P*D + P(c)'*3*D
When it should be:
Weapon 1: A(d) = P*D + P(c)*D
Weapon 2: A(d)' = P*D + P(c)'*2*D
Since the 2x crit means doubling the damage on a crit and 3x means tripling it. The way I had it originally was tripling the damage on the first weapon and quadrupling the damage on the second weapon.
Since all but the difference factored out though, the actual final result was accurate. Still it bugs me I made an error like that.
Heh, wrong again... sigh, what I get for trying to derive equations while on conference calls...
The crit calculation does affect the whole equation:
Weapon 1: A(d) = P*D + P(c)*D
Weapon 2: A(d)' = P*D + P(c)'*2*D
So A(d) - A(d)' = P(c)*D - P(c)'*2*D
Or A(d) - A(d)' = D*(P(c) - P(c)'*2)
We know that P(c) = 2*P(c)'
So A(d) - A(d)' = D*(2*P(c)' - 2*P(c)')
Or A(d) - A(d)' = 0
So you were right Rynjin, they are the same.
Or else I'm going to have to actually do this without being distracted...
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The crit calculation does affect the whole equation
So you were right Rynjin, they are the same.
Yep. What's even more interesting is when you calculate how much the criticals add, compared to the expected DPR.
i.e. starting with the base DPR when criticals never confirm, how much (relative to that DPR) gets added by an x2 crit weapon, how much by an x3 weapon (which is the same amount as a 19-20/x2 weapon), etc.
It's a very simple ratio, which might surprise you.
I stated it before:
Average damage per attack roll = DPH * PH * (1 + CR * CM)
DPH = Average damage per hit
PH = Probability to hit
CR = Critical range in %
CM = Critical multiplier - 1
For a weapon that crits on 20 x3, CR=5%, CM=2 -->CR*CM = 10%
If it crits on 19-20 x2: CR = 10%, CM = 1--> CR*CM = 10%
CR*CM is the average damage that the critical adds.
Adding the keen property to both weapons:
17-20 x2 --> CR*CM = 20% * 1 = 20%
29-20 x3 --> CR*CM = 10% * 2 = 20%
+1 = +5% Damage is a perfectly accurate statement. It refers to average damage done, not damage in any particular scenario.
It all depends on what you mean by "+5%".
+1 = +5% Damage is a perfectly accurate statement. It refers to average damage done, not damage in any particular scenario.
Actually this is almost never accurate. As I showed above if you do 10 damage on an average hit and hit 50% of the time, then in 100 attacks you will do 500 damge. If you get a +1 to your attack in 100 attacks you will do 550 damage. Virtually anyone who is asked would call that a 10% increase in damage. Because it is.
It all depends on what you mean by "+5%".
An additional 5 (the number after 4 but before 6) percentage points (that is, 5 out of 100) added to the base number used for comparison.
It all depends on what you mean by "+5%".An additional 5 (the number after 4 but before 6) percentage points (that is, 5 out of 100) added to the base number used for comparison.
My point was that while an increase of 5 percentage points (e.g. 50% chance to hit becomes a 55% chance) is different than a 5 percent increase (e.g. 100 damage becomes 105), it is rarely clear which of the two is meant by "+5%".
Just call me a smartass and move on please.
A +1 to hit is not +5% damage: as many people has stated before, if your chances were 5% before applying that +1, a +1 to hit means +100% on average or expected damage per attack roll.
A +1 to hit is +5% damage: add 5% of your average damage ___per hit___ to your expected damage ____per attack roll_____
"+1 = +5% Damage is a perfectly accurate statement. It refers to average damage done, not damage in any particular scenario."
The truth is that +5% is below the _lowest_ DPR increase for ACs that you can hit without a natural 20. It has been proved that if you have 5% chances of hitting, the DPR increases by 100% adding a +1 bonus (unless you still need a natural 20 to hit).
If you have 90% chances before adding the +1 bonus, your DPR will increase by .95/.9 = 5.5. That's not taking into account criticals, which add some more.
Tabulated increase in DPR for a +1 bonus (without criticals):
% to hit before bonus / % to hit after bonus / % increase in DPR
90% 95% 5,56%
85% 90% 5,88%
80% 85% 6,25%
75% 80% 6,67%
70% 75% 7,14%
65% 70% 7,69%
60% 65% 8,33%
55% 60% 9,09%
50% 55% 10,00%
45% 50% 11,11%
40% 45% 12,50%
35% 40% 14,29%
30% 35% 16,67%
25% 30% 20,00%
20% 25% 25,00%
15% 20% 33,33%
10% 15% 50,00%
5% 10% 100,00%
Lowest DPR increase: 5,56%
Average DPR increase: 19,42%
Median DPR increase: 10,56%
Highers DPR increase: 100%
So, for any random scenario we can say that DPR increase should be roughly 20%.
But let's take an example. I have a fighter in my campaign with a +6 bonus to attack. If if fights enemies with AC 17, he is expected to have hit 10 times after 20 rounds. Adding a +1 bonus he would hit 11 times. That's 10% increase (11/10 -1)
The oracle's got a +3 bonus. He's hitting that same foe 7 times after 20 rounds. With a +1 bonus he would hit 8 times. Thats 14,3% increase.