Dice statistics and criticals

[Calen]

I was doing some thought experiments on balancing average dice rolls with different dice.
I started wondering: what is the value of a critical role?
Is my math correct?
A d6 has an average roll of 3.5
A d20 critical effect occurs 5% of the time. Or if I expand the crit range it occurs an extra 5% of the time.
So if I roll another dice when I get a critical effect would my average damage be Average{Dice} + (Average{Dice}*.05)?

Thanks,
Not a math nerd

[Jervis]

I was doing some thought experiments on balancing average dice rolls with different dice.
I started wondering: what is the value of a critical role?
Is my math correct?
A d6 has an average roll of 3.5
A d20 critical effect occurs 5% of the time. Or if I expand the crit range it occurs an extra 5% of the time.
So if I roll another dice when I get a critical effect would my average damage be Average{Dice} + (Average{Dice}*.05)?

Thanks,
Not a math nerd

What are you asking exactly? A 20 critIng on a 19-20 is a 10%. Advantage on a roll critIng on a 20 is 9.75%. I recommend the website anydice, it’s how’s the odds of any given roll happening

[Calen]

I recommend the website anydice, it’s how’s the odds of any given roll happening
I looked there and was not able to see what I wanted to know.
What I am looking for:
If there is a 5% or 10% or whatever chance of something happening (like a critical strike with a weapon)
Then what bonus damage is dealt on average?

So if I had a weapon that dealt +2 damage on average and I had another weapon that dealt a crit when a 19 was rolled would that be roughly equal? Would it need to be a 18-20?

[Anonymouswizard]

I was doing some thought experiments on balancing average dice rolls with different dice.
I started wondering: what is the value of a critical role?
Is my math correct?
A d6 has an average roll of 3.5
A d20 critical effect occurs 5% of the time. Or if I expand the crit range it occurs an extra 5% of the time.
So if I roll another dice when I get a critical effect would my average damage be Average{Dice} + (Average{Dice}*.05)?

Thanks,
Not a math nerd

In the following calculations 'd' stands for the number of sides the damage die has, 'h' is how many sides of the die will let you hit a particular enemy, 'r' is the width of your critical threat range and 'c' is your critical multiplier

Okay, so in D&D your average damage on a normal hit is d/2+0.5+modifiers. But you don't t
hit all the time, so we need to factor your hit rate in. In total your average damage against a particular foe without taking criticals into account is h(d/2+0.5+modifiers)/20. So if I'm using a normal longsword, have a +3 Strength bonus, and need a 7 to hit I deal an average of 14(8/2+0.5+3)/20=7(7.5)/10=5.25 damage every round.

Crits complicate this a bit. In most editions crits do double damage, and I could go on about this mathematically, but the simple formula you end up with is:

(h+r)(d/2+0.5+modifiers)/20

If move away from the double damage assumption we have to multiply the critical threat rant, by the critical multiplier minus one. This gives us the slightly less wieldy:

(h+r(c-1))(d/2+0.5+modifiers)/20

Assuming the same hit rate with a 3.X longsword we now average a nice 6DPA (Damage Per Attack).

Bare in mind that this assumes that every critical hits. There's probably a way to take into account cases where that isn't true*, but the effort required to look up the mathematical notation for 'floor' probably isn't worth it. Assume that all 20 sides of the die give you a hit and we get the same results as your equation.

* h=1, r>1

EDIT: if just looking for the excess damage, as long as h>r you get:

r(c-1)(d/2+0.5+modifiers)/20

[MoiMagnus]

I was doing some thought experiments on balancing average dice rolls with different dice.
I started wondering: what is the value of a critical role?
Is my math correct?
A d6 has an average roll of 3.5
A d20 critical effect occurs 5% of the time. Or if I expand the crit range it occurs an extra 5% of the time.
So if I roll another dice when I get a critical effect would my average damage be Average{Dice} + (Average{Dice}*.05)?

Thanks,
Not a math nerd

You're forgetting the probability of hitting the target, but this final result "it increases the average damage by Average{Dice}*5%" is still correct.

Let's take a practical example:
+5 to hit with 1d6+3 damage VS 15 AC
So you hit with 10 on the d20, so 55% chance, which mean
+ An average of 50%*(3.5+3)+5%*(7+3) = 3.75 damages with normal crit
+ An average of 45%*(3.5+3)+10%*(7+3) = 3.925 damages with extended crit

=> An absolute increase of 0.175 damage (which is Average{Dice}*5%), which is a relative increase of 4.7% (this % depends on a lot of things)

[Saintheart]

Maat Mons did some calculations a while back, and I tabulated them:




Threat Range
x2
x3
x4
x5
x6
x7
x8


20
+5%
+10%
+15%
+20%
+25%
+30%
+35%


19-20
+10%
+20%
+30%
+40%
+50%
+60%
+70%


18-20
+15%
+30%
+45%
+60%
+75%
+90%
+105%


17-20
+20%
+40%
+60%
+80%
+100%
+120%
+140%



16-20
+25%
+50%
+75%
+100%
+125%
+150%
+175%



15-20
+30%
+60%
+90%
+120%
+150%
+180%
+210%



14-20
+35%
+70%
+105%
+140%
+175%
+210%
+245%


13-20
+40%
+80%
+120%
+160%
+200%
+240%
+280%


12-20
+45%
+90%
+135%
+180%
+225%
+270%
+315%


11-20
+50%
+100%
+150%
+200%
+250%
+300%
+350%


10-20
+55%
+110%
+165%
+220%
+275%
+330%
+385%


9-20
+60%
+120%
+180%
+240%
+300%
+360%
+420%


8-20
+65%
+130%
+195%
+260%
+325%
+390%
+455%


7-20
+70%
+140%
+210%
+280%
+350%
+420%
+490%


6-20
+75%
+150%
+225%
+300%
+375%
+450%
+525%


5-20
+80%
+160%
+240%
+320%
+400%
+480%
+560%


4-20
+85%
+170%
+255%
+340%
+425%
+510%
+595%


3-20
+90%
+180%
+270%
+360%
+450%
+540%
+630%


2-20
+95%
+190%
+285%
+380%
+475%
+570%
+665%



These are the expected bonus damage percentages for a given threat range and/or threat multiplier over the course of a campaign, assuming a natural 1 is always a miss and a natural 20 is always at least a critical threat. (IMPORTANT EDIT: Just to clarify, these percentages assume you're playing third edition, which has particular rules about critical hits.)

Maat's calculations were based on the following math:

To get the % increase, we need to take (a - b) / b, where a is the damage with critical hits, and b is the damage without.

So, to involve way too much math, we're going to define the following variables:
Phit = probability that attack roll is high enough to hit target's AC
Pthreat = probability that attack roll is within weapon's critical threat range
D = weapon damage before critical multipliers
M = critical multiplier

So, what's the expected value for damage against crit-immune enemies? That's pretty straightforward.
Phit * D
That's the probability that we hit, times the damage we deal if we hit
That's our b from (a - b) / b.

What's the expected value for damage against enemies that are subject to critical hits? Slightly more complicated.
It's the odds that we critically hit times the damage of a critical hit, plus the odds of a non-critical hit times the damage of a non-critical hit.

The critical hit part of that is pretty straightforward. Assuming that any roll good enough to be a critical threat is also good enough to hit, it's
Pthreat * Phit * D * M

There are two ways of getting a non-critical hit. We can roll a critical threat and fail to confirm, or we can roll a hit that wasn't a high enough roll to be a threat. So the odds of a non-critical hit is the sum of the odds of those two events.
Pthreat * (1 - Phit) + Phit - Pthreat
And then of course multiply by non-critical damage, D.

So the big, complicated formula becomes:
[ Pthreat * Phit * D * M + (Pthreat * (1 - Phit) + Phit - Pthreat) * D - Phit * D ] / Phit * D
Which, when you cancel out the terms, becomes the much friendlier-looking formula:
Pthreat * (M - 1)

So, and this is an important result, once both the initial attack roll and the critical confirmation roll are both factored in, they both factor out.

As such, we can very easily calculate the value of the critical threat range/multiplier of weapons:
x3: +10% damage
19-20/x2: +10% damage
x4: +15% damage
18-20/x2: +15% damage
19-20/x3: +20% damage
17-20/x2: +20% damage
19-20/x4: +30% damage
15-20/x2: +30% damage

[Satinavian]

I looked there and was not able to see what I wanted to know.
What I am looking for:
If there is a 5% or 10% or whatever chance of something happening (like a critical strike with a weapon)
Then what bonus damage is dealt on average?

So if I had a weapon that dealt +2 damage on average and I had another weapon that dealt a crit when a 19 was rolled would that be roughly equal? Would it need to be a 18-20?
That is impossible to answer without knowing hit bonus and armor class (and DR even more, if existing).

The additional critical damage is fixed but the expected base damage obviously depends on your chance to hit at all so there is no way around those situational circumstances when you want to compare the two.

An extra point crit range is roughly comparable to an extra point to hit. But even here maneuvers like power attack, DR, weapon enchantments that only triffer on crit or don't get multiplied and possible ways to boost crit multiplier can make those very different in detail.


Traditionally it seems that a point of crit range was seen somewhat equivalent to one or two points damage. But as it allows other synergies and is useful for different builds, it can work out way differently in practice.

[Tanarii]

There are two ways of getting a non-critical hit. We can roll a critical threat and fail to confirm, or we can roll a hit that wasn't a high enough roll to be a threat. So the odds of a non-critical hit is the sum of the odds of those two events.
Pthreat * (1 - Phit) + Phit - Pthreat
And then of course multiply by non-critical damage, D.

So the big, complicated formula becomes:
[ Pthreat * Phit * D * M + (Pthreat * (1 - Phit) + Phit - Pthreat) * D - Phit * D ] / Phit * D
Which, when you cancel out the terms, becomes the much friendlier-looking formula:
Pthreat * (M - 1)

So, and this is an important result, once both the initial attack roll and the critical confirmation roll are both factored in, they both factor out.Very interesting 3e analysis with critical threat confirmation. It's interesting that it greatly simplies down in that case to remove the intitial attack roll.

For 5e with no confirmation roll, this section gets modified to:

Normal hit: (Phit - Pthreat)*D

Big complicated formula: [ Pthreat * Phit * D * M + (Phit - Pthreat) * D - Phit * D ] / Phit * D

Cancel out terms: [ Pthreat * Phit * D * M - Pthreat * D ] / Phit * D
Becomes:
Pthreat * (M - [1/Phit])

So the original hit chance is very relevant in that case.

[Calen]

Well that is certainly a wealth of information to glean through. Thanks everyone.

[Psyren]

It's specific to D&D 5e, but I highly, highly, highly recommend LudicSavant and AureusFulgen's DPR Calculator. (https://docs.google.com/spreadsheets/d/14WlZE_UKwn3Vhv4i8ewVOc-f2-A7tMW_VRum_p3YNHQ/edit) It lets you put in parameters such as your attack bonus, number of attacks, the target's expected AC (you can find a table of these values by CR in the Dungeon Master's Guide, page 273), chance to crit, and even bonus stuff that happens when you crit like a Barbarian's Brutal Critical, and use all of that to figure out what your build's average DPR will be at any given level. They have detailed explanations of the underlying math if you're into that, but you don't need that kind of knowledge to be able to make use of it.

There are even bells and whistles such as telling you what your average DPR will be if you also have advantage (or disadvantage) on the hit, what monster AC breakpoints you should be using power-attack style abilities and when you shouldn't, and what the effects of common buffs like Bless or Bardic Inspiration that add additional dice will be.

Other games might have similar tools but this is the best one I've seen so far.