# 3d6 vs a d20: What is the effect of a different probability curve?

What is the effect on game play of replacing d20 rolls with 3d6 rolls in a D&D 4e campaign?

I started thinking about this after reading this question where the example was given of playing a d20 system with 3d8 (actually 3z8, but I digress). I like math and probability and it got me curious...

After some searching, I found out that in D&D3.5's Unearthed Arcana, there was a rule variant which replaced every d20 roll with 3d6. Several things had to change (for example, a critical threat range of 19-20 needs to be shifted down because you can't roll a 19), but since the average result was the same (10.5), the math and the mechanics didn't need to change much. What changed was hard things got very hard and easy things got very easy. Beating a DC 17 under d20 happens 20% of the time. Under 3d6, it happens < 2% of the time. Likewise a DC 6 goes from 75% success to a 95% success rate.

From reading what I could find, this variant supposedly makes for a 'grittier' game. The average happens most of the time and that favors the stronger side, which is usually the players. But great rolls don't happen very often.

Here are my questions:

• Are there any mechanics / stats changes that are needed to keep things balanced?
• How does it affect tactics (both for combat and skill challenges)?
• Does rolling average results most of the time turn it into a 'grinding' game that takes forever?
• Does it ever fall apart?
• Is it still fun?

Off the bat, I would guess that Aid Another becomes an action that characters would take on a frequent basis. If the fighter needed a 15 to hit the big guy, under d20, two people aiding her would increase her chances from 25% to 45%, maybe not worth it. Under 3d6, their aiding would change it from 9% to 50% and adding a third aider would up that to 75%.

Here's a table of rolling N using 3d6 for reference:

``````N      =N      <=N    >=N
``````
```3    0.46%   0.46% 100.00%
4    1.39%   1.85%  99.54%
5    2.78%   4.63%  98.15%
6    4.63%   9.26%  95.37%
7    6.94%  16.20%  90.74%
8    9.72%  25.93%  83.80%
9   11.57%  37.50%  74.07%
10  12.50%  50.00%  62.50%
11  12.50%  62.50%  50.00%
12  11.57%  74.07%  37.50%
13   9.72%  83.80%  25.93%
14   6.94%  90.74%  16.20%
15   4.63%  95.37%   9.26%
16   2.78%  98.15%   4.63%
17   1.39%  99.54%   1.85%
18   0.46% 100.00%   0.46%
```

(produced using this calculator)

-
Hmmm ... if it's only to get a different distribution, it might be easier rule-wise to just use 2d6+1d8 (that 20, right?) Would also make an interesting distribution! :-) –  Martin Oct 28 '10 at 20:01
Did you try this? I did run dnd3 with 3M20 (see below answer) and it did work ok but it is biased towards the players. –  David Allan Finch Nov 2 '10 at 11:49

A good way to analyze the differences between the two distributions is to imagine a head-to-head contest between characters.

First, suppose you have two identical characters, `A` and `B`, rolling off against each other with d20. They tie 5% of the time; 47.5% of the time one wins; 47.5% of the time the other wins. In contrast, if you use 3d6, ties occur 9.2% of the time and each wins 45.4% of the time. Not a huge deal. Let's discard the ties and just concentrate on who wins more, `A` or `B`. Now let's start giving them bonuses. Since we haven't said who is whom, we'll just declare that `A` is the stronger one and `B` is the weaker one.

``````A's bonus  3d6                     d20                    3d6 ratio
=========  =====================   =====================    over
=========  A-wins  B-wins  ratio   A-wins  B-wins  ratio  d20 ratio
---------  ------  ------  -----   ------  ------  -----  ---------
+0         45.36%  45.36%    1.0   47.50%  47.50%    1.0      1.0
+1         54.64%  36.31%    1.5   52.50%  42.75%    1.2      1.2
+2         63.69%  27.94%    2.3   57.25%  38.25%    1.5      1.5
+3         72.06%  20.58%    3.5   61.75%  34.00%    1.8      1.9
+4         79.42%  14.46%    5.5   66.00%  30.00%    2.2      2.5
+5         85.54%   9.65%    8.9   70.00%  26.25%    2.7      3.3
+6         90.35%   6.08%   14.9   73.75%  22.75%    3.2      4.6
+7         93.92%   3.59%   26.2   77.25%  19.50%    4.0      6.6
+8         96.41%   1.97%   49.0   80.50%  16.50%    4.9     10.0
+9         98.03%   0.99%   99.0   83.50%  13.75%    6.1     16.3
+10        99.01%   0.45%  220.0   86.25%  11.25%    7.7     28.7
+11        99.55%   0.18%  552.9   88.75%   9.00%    9.9     56.1
+12        99.82%   0.06%  1663    91.00%   7.00%   13.0    127.9
+13        99.94%   0.02%  6661    93.00%   5.25%   17.7    376.0
+14        99.98%   0.00% 46649    94.75%   3.75%   25.3   1846.3
``````

Okay, so what does this tell us?

First, we can see that with big bonuses, `A` slaughters `B` head-to-head in rolls in 3d6, whereas with d20 the benefit that `A` gets over `B` is pretty modest (has to get all the way up to +11 before `A` is tenfold more likely to win than `B`!).

But, second, if you look at the ratio of ratios (that is, how much advantage `A` vs `B` has in 3d6 compared to `A` vs `B` in d20), you find that in `3d6` the bonus is pretty much squared compared to d20 (low values only--then it gets way, way more extreme later on).

So, what does this mean? Well, basically, if under 3d6 you have a +1 bonus more than someone else, it feels like a +2 difference in d20. +7 feels like +14.

So the concise explanation is: moving from d20 to 3d6 amplifies differences, making them feel about twice as large as before. (Of course, almost nothing is actually resolved as a head-to-head test, but it's a useful thought experiment.) You can cleave through hordes of lesser beings with that much more ease, and your betters become that much more fearsome. In fact, better just stay away from them. There are some kobolds that need slaying. Right? Right.

-
I had an instinctive feel from looking at the results, but this first time I have seen it so well put. Thanks. –  David Allan Finch May 6 '11 at 8:23
Excellently put. Why doesn't this have more up votes? –  Pureferret Dec 11 '11 at 13:55
@Pureferret - Glad you found it useful. It probably has few votes in part because it was a late answer--the question was asked in Sept. 2010, and I didn't notice it and write an answer until over 6 months later. –  Ichoran Dec 12 '11 at 13:52
@RobertF - That's not how 3d6 works--you gradually get better at first, then accelerate away increasingly fast. d20 accelerates away also, but not as fast. Exploding d20s would be even slower (especially if they exploded on 18 or so). –  Ichoran Feb 1 '14 at 21:43
@RobertF - Try fighting a hundred opponents at once and you'll notice a massive difference between 96.4% and 99.98%. –  Ichoran Feb 2 '14 at 16:03

First off, those little +1s and +2s are going to be much more important. Being flanked is suddenly a matter of, say, a 50% increase in their chance to hit you rather than a 10% increase. You noticed this with Aid Another, but it'll come up other places as well. Any power that forces an enemy to grant combat advantage becomes much, much more powerful. Being dazed is traumatic.

I think the grind would decrease. The base math aims towards characters hitting on a 10 or better; that becomes a 62% chance rather than a 55% chance, so damage output will rise. Optimized characters who're hitting on a 9 or better become really deadly as opposed to pretty deadly. Again, scraping out an extra +1 to hit matters a ton.

You won't be able to throw higher level monsters at parties as easily, and lower level monsters will become less threatening. The level band of reasonable opponents narrows, because things that were sort of hard to hit become really hard to hit. I think this is the biggest argument against the change, personally. Say you needed a 14 or better to hit a monster; OK, that's 35%, not too bad. But 16% is significantly more demoralizing.

Powers and abilities that trigger on a critical become much less worthwhile unless you move the default crit range to 16+. I suspect you'd almost have to make that change.

-

Bryant is right about the Bonuses. In GURPS I had be to be careful about bonuses as beyond a certain point success (or failures) is all but certain. I recently returned to D&D, playing Swords & Wizardry, and one thing I noticed over GURPS is how more variable the the results seems. With the d20 numbers were all over the place and even character with high bonus can have what bad streaks.

In contrast with GURPS the bull curve meant that once your skill was pushed beyond the 12 to 13 chance of success (roll low) you feel more competent as you rolled more 9's, 10's, 11's, and 12's more than other results. There was a less swing in the numbers if that make sense.

It made combat a little more predictable as you were mostly getting average results. So you could plan accordingly.

Another interesting variant I been seeing is 2d12 for 2 to 24.

-
I like 2d12 -- it's a gentler curve while still getting some of the same effect. –  Bryant Sep 14 '10 at 13:50
I particularly like the predictability when it comes to skill checks (as those don't typically have as many rolls to establish an average as attacks do). It's nice to be able to rely on the things you're supposed to be good at. –  AceCalhoon Sep 14 '10 at 14:43
@AceCalhoon Yeah, I have always found the D&D skill checks far too variable. I have seriously considered 3d6 or perhaps even 4d6-4 (gives 0-20 instead of 1-20) for non-combat actions. –  Loren Pechtel Aug 9 '14 at 2:47

Using 3dX was a fun idea for me more because of skills than combat; everyone's perspectives feel very combat-oriented to me. (bear in mind, I have a 3.5 mindset)

The problem I had which was solved with 3dX, was that some people had super cool character concepts which were just impossible or unsatisfying with a d20... I can't explain the effect on probability but I can explain the effect on Player satisfaction by showcasing the problem I had...

All numeric values are approximations as I am lazy.

## The Problem

Imagine you're playing a 5th level Rogue. You've come up with a solid burglar-type character concept which you feel passionate about, and you really want this Rogue to be a top notch entry guy. You've maxed your Open Lock skill at 8 ranks - 1/10th of your total skill points - with a +2 from Dex. Yet, despite this significant investment in a pretty integral part of your character, you still have a 50-ish% of failing to pick a simple lock. Not even a special lock; a simple \$%^&ing lock.

Meanwhile, BamBam the Barbarian is being played by Joe the Casual Gamer, and the character concept consists of hitting things and hitting them hard. He's become very good at it; he sure hasn't wasted 5 levels to only have a 50% chance of hitting a simple enemy. He's having an awesome time with his character becoming everything he envisioned.

You get bitter and lose your passion about the character, and the game.

## How can this problem be solved?

### Option 1 - Make Open Lock DCs Lower

What if we lower the DC so picking a simple lock is DC15?

This is not a good solution; now our Rogue is satisfied - he's an entry guy with a 75% chance of picking a simple lock - but next level BamBam tosses all his skill points into Open Lock; with his +4 Dex mod (Dex is important to BBNs) he has a total of +6... with a 60% chance of picking a simple lock, he's not a half bad entry guy either!

So, now everyone can be an entry guy in just one level! (and when eveyone's super...)

### Option 2 - Use 3d6

Using a DC of 20 and a 3d6 roll, the difference between BamBam's +6 and the Rogue's +10 translates to something like a 50% difference in their outcomes to pick a simple lock; the Rogue will succeed something like 70% of the time and BamBam only succeeds 20ish% of the time.

So in this instance, we can see a player with a cool character concept who is able to follow through on that concept as a result of using 3d6 instead of 1d20.

-

A better solution is to use Mid 3d20 (3M20). That is to select the middle roll from three d20. This has the advantage of creating a parabolic curve (*) but still giving you the full range of a d20.

``````The probs are:

mid20    Prob % of
TN    Prob    Eq or higher
%
1    0.725    100
2    2.075    99.275
3    3.275    97.2
4    4.325    93.925
5    5.225    89.6
6    5.975    84.375
7    6.575    78.4
8    7.025    71.825
9    7.325    64.8
10    7.475    57.475
11    7.475    50
12    7.325    42.525
13    7.025    35.2
14    6.575    28.175
15    5.975    21.6
16    5.225    15.625
17    4.325    10.4
18    3.275    6.075
19    2.075    2.8
20    0.725    0.725

Mid of 3d20

Value    4.0 bars per %
1 :  |||
2 :  ||||||||
3 :  |||||||||||||
4 :  |||||||||||||||||
5 :  |||||||||||||||||||||
6 :  ||||||||||||||||||||||||
7 :  ||||||||||||||||||||||||||
8 :  ||||||||||||||||||||||||||||
9 :  |||||||||||||||||||||||||||||
10 :  ||||||||||||||||||||||||||||||
11 :  ||||||||||||||||||||||||||||||
12 :  |||||||||||||||||||||||||||||
13 :  ||||||||||||||||||||||||||||
14 :  ||||||||||||||||||||||||||
15 :  ||||||||||||||||||||||||
16 :  |||||||||||||||||||||
17 :  |||||||||||||||||
18 :  |||||||||||||
19 :  ||||||||
20 :  |||
``````

Source: rpg-create

For threats, crits and fumbles there are several options. The simplest is to just say that the threat range is now the crit range; this reduces the chance of a crit on average but is quick to work out. The second is to say that 18 or greater is a '20' on a threat roll and 17 or greater is a '19-20', 16 is '18-20', 15 is '17-20'. After that the probs no longer work correctly

There are some skills where failure chance is -10 or -5 on the DC, you might want to work out the probability of fail for a average character at the level and adjust it using the above 3M20 roll.

NOTE: all hill probability curve systems (*) are biased towards the players as average result are more useful to them in the long run.

NOTE 2: (*) Originally this said bell curves but as has been pointed out a standard 3d6 gives a more correct bell curve and the 3m20 gives a parabolic curve. see 2m20 vs 3d6 The original points are still basically true just that low / high numbers are a bit more likely that using 3d6 (which might not be a bad thing).

-
That's a curve, but not a bell curve. With a ~1-20 bell curve, you'd expect 16 to happen a third as often as 11, and 20 would be ~two in a thousand. –  TBohne Aug 30 '14 at 10:51
@MooingDuck - Good point. I had not spotted that before. I will update the answer, thanks. –  David Allan Finch Sep 1 '14 at 9:14

Yeah, 3d6 as a mechanic in general is good - GURPS does it - but the main problem is that D&D has a pretty wide range of bonuses. It makes someone with a +2 edge over someone else way better. Having said that, I don't mind that, and have messed around with 2d10 as a halfway step.

But if you are really into the rules, there's a lot of problems it brings up. Crits have to be completely different of course; I'd redefine crits as "beating the target number by 5".

In general I don't mind things being more normalized and a level 5 guy being way better than a level 3 guy. And it has the advantage of single bosses no longer being meat for a PC party. But "little" bonuses become perhaps too much. I'd switch Aid Another to +1, for instance. And a fifth level character can have an attack bonus from like +3 to +9, which means those who aren't minmaxed out the ass on Strength etc. are going to die to those that do. (I am speaking from a D&D3 viewpoint, I don't play 4e, but I assume it has the same syndrome.)

I think it works best if you were planning on a pretty tight level band anyway. Like I enjoy low level games; the PCs in the game I run are level 4 after a year of 7-hour every-other-week sessions. The tighter band would work well for that. If you're planning on cranking through levels 1-30 it won't work as well.

One thing I've messed with to make this work - and to minimize the horrible swinginess and min-maxing in D&D - is to make a maximum bonus. It seems silly to me that someone can stack +20 in bonuses onto what is otherwise a +2 base attack bonus, so I cap it. You can only ever double your bonus with all combinations of strength/magic/synergy/whatever. (Doing the same to damage really helps too.) Having more bonus is still good because it can help you overcome penalties...

-
with 4e its not so bad since attack and damage bonuses are tied to each class' primary stat, so everyone has +3 to +5 accuracy and damage at level 1. And they all scale togeather, so its +21 to +25 at level 30. –  TBohne Aug 30 '14 at 10:55

3d6 does a Gauss distribution, this is why the probability isn't equal with the linear distribution of 1k20 (it is also a Gauss distribution, but it's slope is exactly zero). If you take a look at the distribution pictures, you'll understand everything. :) Even without strong mathematical skills one can see the more dice used the more peaky the curve is.

Just comparing the graphs, the odds to roll 6-15 are equal or better than the odds on a d20 (and so 3-5 and 16-18 are less likely than on a d20), and the range 8-13 is more than twice as likely to be rolled than on a d20.

-

I think just looking at the bare percentages underestimates the impact of a +1 bonus. If a roll requires an 18 or greater to be successful, a +1 bonus under d20 increases the chances from .15 to .2 - but this is an increase of only 33%. Under 3d6, it increases the chances from a feeble .0046 to .0185 - but this is an increase of 300%. (I'm intentionally ignoring "crits always hit" mechanics, here.) A character who needs an 18 to hit and gets a +1 bonus to hit deals 33% more damage under d20, and four times as much damage under 3d6. The relative increase is always better for 3d6 all the way down to rolls that require 8 or greater to succeed. It's only when you're looking at rolls that are almost a sure thing (90% chance to succeed before the bonus) that the bonus becomes relatively unimportant, and that's largely because rolling 7 or better is almost a sure thing under 3d6, whereas it's still only 70% likely under d20.

Similarly, under 3d6, a +5 bonus becomes absurd. If an 18 or greater is needed to hit, a +5 bonus does 56 times as much damage, compared to 6 times as much damage under d20. It's much better, again, until it reaches the point where it almost guarantees success under 3d6.

Under 3d6, if you're fighting something that you can hardly hit, you're likely going to be better off spending three potential attacks trying to earn a +2 bonus against it than you would to just swing four times. A +1 bonus is worth giving up a potential attack as long as you need a 16 to hit; a +2 bonus is worth giving up a potential attack if you need a 13 or better to hit.

So. The overall damage against creatures that require more than an 11 to hit (after innate bonuses) is going to go down sharply, and the overall benefit of spending turns trying to improve chances to hit is going to go up sharply against those same creatures. A character that hits only on a 16 or better is essentially worthless for damage except to the extent they can provide bonuses to other players.

But most importantly, differences in hit chances around the center are going to be greatly magnified. Under d20, a PC who hits on a 13 or better and averages 8 points of damage per hit will have the same damage output as an NPC who hits on a 9 or better and averages 5 points of damage per hit. Under 3d6, the low-damage NPC is going to do nearly twice as much damage as the higher damage PC, because it's going to hit so much more often. As Bryant mentions, it makes the "balanced encounter" sweet spot much smaller. From one perspective, that's good - many encounters are likely to either go quickly bad or quickly good. But winning a tough encounter on a series of lucky rolls goes out the window.

-

I think it's really important to note that GURPS (which is 3d6) doesn't have the concept of 'armor class' in the same way D&D does. The target number may go up by 2 or 3 points, at most, due to the armor the target character is wearing. Most of the armor's protection is essentially DR. There's no dex bonus to defense.

Instead, the defender makes a separate roll (at least in 3rd Ed, not sure about 4th) to see if they can parry or dodge the attack, using their own defense skill total (which is usually pretty low). This makes the modifiers and difficulty totals less key.

I enjoy the predictability of GURPS skill checks more than the linear randomness of D&D. I think it's a matter of personal taste. I also think diminishing returns make a lot of sense. In D&D, your total just keeps going up; it's difficult to set a reasonable skill check DC when you have a character in the party with a +6 bonus to a poorly-trained skill, and another character with +20. Similar difficulties apply to setting a monster's AC, though even low-BAB characters do increase their hit totals as they go up in level. The GURPS model provides a slightly more level playing field for powerful vs. inexperienced characters.

In terms of balancing the game, you need to be cognizant of the difference when changing the system, and designing opponents. The bell-curve probability distribution requires a less heavy-handed approach to setting target numbers, and making a tough enemy harder to hit takes a bit more fine-tuning. I think the GURPS solution of using a separate check is a good one.

To convert 4E to GURPS, you would really have to go over each monster's statistics, figure out the probability of an appropriate-level character hitting (or being hit), and translate those numbers into reasonable small modifiers, and defense check values. The result will be that lower-level characters are much more likely to hit higher-level monsters, and higher-level monsters will dodge attacks from higher-level characters more frequently.

It might be easier to simply use GURPS, and re-create monsters and characters in the new system. It's a fairly dramatic game balance change, and won't work if it isn't well-thought-out.

-
Just grab GURPS Lite and the first couple of items in the GURPS Dungeon Fantasy series for a quick, low-cost game, or grab the core books, Fantasy, Banestorm, and Magic for the full-on GURPS version. Oh, I'd probably pick up Low-Tech, too. –  gomad Mar 18 '11 at 21:19

Something I just realized about the relative value of a +1 or +2 on a curve versus on a linear scale is that it has different effects based on where you limit things. If you're taking a system like E6, where you stop leveling up in most ways and have a skill rank cap, then you'd want each +1 to be worth more. Moving to a 3d6 curve might mess things up in Mutants & Masterminds, but if you're going for something intended for a lower power level (not unlike where GURPS is strongest) then it can be quite beneficial. Differences in power level are also much more pronounced. If you wanted a martial arts themed game where differences in belt rank really mean something as far as ability goes (which has nothing to do with reality!) using a 3d6 system would mean that a higher level character would win in opposed checks with far greater likelihood than with a d20.

It's rather the same as assuming everyone is taking 10 all the time. If something's impossible to achieve by taking 10, odds are really low that you'll succeed on 3d6. Similarly, if something succeeds every time when taking 10, you barely have to bother rolling, as success is pretty much a matter of fact. You'd end up with each character level being a massive leap rather than a gradual improvement. That could be a good or bad thing, but probably bad if you're trying to balance according to published CRs and ELs.

-

Going from d20 to 3d6 would take some of the drama out of rolling. Everyone is always watching to see if a natural 20 or 1 is rolled -- ready to join the chorus of cheers or groans.

If the d20 were replaced, then I think a normal curve is too low. The average people wave from shops and fields as the adventurer marches off to slay the foes. The hero rolls 4d6 and discards the lowest die. If you spend time training and testing, then you expect to get consistently above-average returns.

Maybe even have an "epic" fudge die, like a d6 marked (1, 1, 1, 2, 2, 3), that is added to the roll to see if you gave it that extra effort and made a 20 to crit.

-
The description I found uses a table that scales down crit ranges to keep their frequency approximately the same (i.e. 19-20 becomes 15-18, both around 10% chance). –  yhw42 Sep 16 '10 at 15:47
If you want to keep the drama whats wrong with keeping fumbles on three 1s and criticals on three 6s? They don't happen as often of course but imo thats better. –  Jagged Oct 19 '10 at 9:04

There is an website called anydice.com that do dice probabilities.

You can find the "at least" problablilities of 1d20 and 3d6, this means the probabilities of rolling certain number or higher. On a 1d20 the chance of rolling at least 1 is 100% and the chance of rolling 3 on a 3d6 is 100%.

An 18 on both doenst mean the same thing, you have 0.46 chance of rolling an 18 or more on 3d6 and but have 15% chance of rolling 18 or more on a 1d20.

If people have have 0 to 100% chance of doing tasks, the 3d6 guy that rolled 18 will complete more tasks than the guy 1d20 guy that rolled 18.

We can use that to gauge how powerfull some roll is, 3 on a 3d6 and 1 on a 1d20 represent the same thing.

By using the "power" of those numbers and comparing them:

Both rolls tie 0.65% of the time.

3d6 Wins 47.55% of the time

1d20 wins 51.81 % of the time.

-
This mostly made sense until "by using the 'power' of those numbers", at which point you completely lost me. Both rolls? Wins? What? –  DCShannon Apr 2 at 17:44
What I meant by power is the chance of rolling at least this number, different dices are just different ways to show probabilities. The power of an number X means how unlikely it is to roll this (or a higher one) number. Because if you rolled an more unlikely roll, this means you should be awarded for that feat and on a contest between player a and b the one that roll the most unlikely must roll win (since they are using different dices) and not the only with the highest value. –  Mirou Apr 6 at 11:51
That explanation doesn't really make sense either (you're quantifying how 'unlikely' stuff is now?), but even if it did, it should be edited into the question. Comments are transient and may be deleted without warning. –  DCShannon Apr 6 at 21:08