Saturday, June 21, 2008

3d6: The Odds

Cubes cubed. A trio of six-siders. Three Dee Six, or Three Die Six. No matter how you slice it, 3d6 is gaming perfection. Some might ague that 3d6’s vastly more popular little brother 2d6, of Craps fame, is gaming perfection, and while I can’t really argue that point, I can say that 3d6 is perhaps more aptly D&D perfection.

Over the past decades, the D&D brand has done a good job of convincing everyone that in fact the d20 is D&D perfection. Sure the twenty-sider is indeed D&D’s famous symbol now. It’s unmistakably “D&D”, and always will be. Let’s be real, though. It’s kinda boring when you get right down to it. Nevertheless, the big bad d20 holds the two most important D&D rolls in it’s chubby little fists; To Hit and Saving Throws. So, I’ll leave Mr. Big Britches d20 alone and focus on my favorite little cubes and the character generation rules so perfectly incorporated into D&D by Mr. Gygax and Mr. Arneson.

What’s so great about 3d6, you might ask. Quite simply, the awesome bell curve it provides. While we are only generating a range of 16 possible numbers, the underlying odds are rather complex. Here then are the actual odds for rolling each and every 3d6 sum, from 3 to 18 (rounded numbers):

3: 0.5% (actually 0.46, or 1 in 216, but rounded off for this table)
4: 1.4%
5: 2.8%
6: 4.6%
7: 6.9%
8: 9.7%
9: 11.6%
10: 12.5%
11: 12.5%
12: 11.6%
13: 9.7%
14: 6.9%
15: 4.6%
16: 2.8%
17: 1.4%
18: 0.5% (as 3’s note above)

Now, maybe you already knew all of this; if so I must tip my hat to your gaming knowledge! Me, I’m just beginning to really absorb it. Rolls of 9-12 account for 48.2% of all rolls, and expanding that range a bit, rolls of 8-13 account for 67.6%, rolls of 7-14 account for 81.4% of all rolls. Only 18.6% of all rolls will be outside of this 7-14 range (an important one in OD&D terms), with half, or 9.3% of all rolls, being 6 or less, or 15 or more, respectively. So, less than a 1 in 10 chance of having superior (15+) or inferior (6-) scores in any given ability. Chances are that one of your six stats will be either superior or inferior.

Let's just stop here a moment and consider that 9-12 range, the 48.2% of all rolls category. The fact that nearly half of all of the rolls fall within this range is pure gaming quintessence. The rules state that this is the average ability score for characters. Lo and behold, this isn't simply conjecture, it's true when using this 3d6 method. Pure genius on the part of Gygax and Arneson. When using this dice rolling convention, we end up with basically one half of all character abilities being lumped together in this average range. Aside from increasing the number of dice (and perhaps making the extreme scores too illogical), is there a better way of randomly determining character ability scores that these two could have devised? I for one think not.

It’s such a grand gaming convention, the process of rolling those innocent 3d6 when creating a character. I enjoy this bell curve so much that I plan on killing characters by the wagon full in my upcoming campaign, just to see the bell curve in action more often. Just kidding! I would like to see the 3d6 roll more often in games, though.

I’d enjoy a RTH or Saving Throw system using these dice, but I think that rolling more than one or two dice all night long might become tiresome. The mini-curve provided by 2d6 has worked well for Craps, and works well in Chainmail and other games. This notion just might inspire me to move forward with my desire to somehow incorporate the Chainmail 2d6 RTH into my OD&D games, but that is indeed a topic for another day.

I prefer this table to just about any other used in D&D, whether it’s a d00 percentage roll, or a d20 RTH/Saving Throw. I’ve crunched up the Solstice RTH, by using my gaming crew’s old tried and true method of rolling a d6 and a d10 together, and I find that the small nuances of that combination introduce a wee bit of definition to the old linear 1-20 rolls.

To paraphrase Mr. Gygax again:

The dice are your tools. Learn to use them properly, and they will serve you well.
The 3d6 bell curve is just begging to be used more. I would welcome ideas or suggestions in this regard.

~Sham, Quixotic Referee


Anonymous said...

You mention how you'd like to see a RTH or Saving Throw system with the d6's, but is that not how the big bad d20 became the boring behemoth he is now today? He gets used one too many times - attacks, saves, ability checks, a number of random tables in the books and modules, etc.

I think part of what makes the 3d6 so special is that they're used for this one-time (or multiple times if you play with a vindictive DM), super special event. We love the 3d6 because of it's uniqueness, and because of it's association with new life.

Sham aka Dave said...

I agree somewhat. I'm sure if I instilled a 2d6 save or RTH system, I might end up waxing nostalgic about the d20.

The 3d6 roll is rather special because it's only used for character creation, true.

I just think the 3d6 bell curve is a great tool for determining other outcomes too. It's a bit unwieldy for the oft rolled tasks like RTH and saves, though.

trollsmyth said...

I love the bell curve as well. In my non-D&D fantasy heartbreaker, I use escalating numbers of smaller dice to represent greater skill. That is, a neophyte might roll a single d20, while an apprentice would roll 2d10, and a master would roll 4d4.

You could do something similar with a skill system for OD&D, but honestly, I think skills are probably not something that should be rolled regularly in D&D.

Want to get some really wacky results? Try one die divided by another. Like 1d20 divided by 1d4. There was an article about this in an old Dragon magazine years ago (like, back in the '80s). I made some magic swords use this method for damage done, and declared that 1 was the lowest result. Usually, the sword did something like 4 or 5 points of damage, but every now and then you could do as much as 20 points of damage on a successful hit.

Anyway, I'd like to see spells use the bell-curve, especially healing magics. It really sucks to roll minimum on a healing potion or a cure light wounds. If you replaced the usual 1d6+1 with a 2d4, I think you'd make the spell even more useful than it is now. I'm also tempted to replace the fighters 1d8 hit dice in Moldvay/Cook with 2d4 for the same reasons.

I'm not sure I'd replace the to-hit roll, however. That makes an average roll far more likely. That means you're far more likely to hit folks with poor ACs, but far less likely to hit foes with good ACs. That makes armour even more useful, and heavily armoured foes far more dangerous than they were before. So no, I don't think I'd touch that one.

- Brian

Sham aka Dave said...

Great comment, Brian.

Try one die divided by another.

I'd never considered this before. That's a great concept, and I like it's possible use for magic weapons.

The thing about the Cure spells, that reminds me of a house rule I used to use, that a Cleric could never cast a heal for a number of points lower than his own level (to the max of the spell, that is). Another I toyed with was +1 point per odd level of the Cleric for heals, but I think I prefer the former. So, in AD&D, a CLW always heals for 8 once that Cleric has reached Level 8. I'll need to incorporate something like that. I always hated seeing a player do a CLW for 1 point.

I like your 2d4 idea for Fighter HP's, that's a very good one.

I'm also hesitant about the 2d6 RTH (which was the Chainmail version, or standard OD&D method). I don't know anyone who actually uses anything besides the "alternative combat system" which became the standard in later editions.

trollsmyth said...


And the more I think about it, the more I like giving dwarves 2d4 or 4d2 hit points per level, just to represent what hardy and dependable little bastards they are. ;)

- Brian

kirel said...

Sorry for adding to an older post like this. I ran across this post in searching for 3d6 odds. Your listing was perfect for me. Thanks.

I've played D&D for many many years now, but have always been bothered with the statistics for d20 rolls. I just wanted to note here that I just started playing GURPS, and was happy to see that the whole system uses 3d6 for everything. A very pleasing difference!

Sham aka Dave said...

I'm glad the post was useful, Kirel.

One of these days I'm going to have to check out GURPS. In many ways it sounds perfect for much of what I try to do in tinkering with OD&D.

Jake said...

Just came across this post. Thank you.

I love the 3d6 (and the 2d6) bell curve. (I really started to notice the beauty of the 2d6 curve when playing 3E - the damage output from the greatsword (2d6) really draws attention to it when played alongside a longsword (1d8).

Apparently someone at Green Ronin is a fan of the 3d6 bell curve as well. I hear the new Dragon Age RPG is based on 3d6 and their Song of Ice and Fire RPG is also based on d6s (the number of d6s rolled is based on your level of expertise in a given "ability).

Sham aka Dave said...

Thanks, Jake. I've not checked out any Green Ronin publications. I just might need to look into the Dragon Age RPG though. Thank you for the info!

Anonymous said...

Me and my Dad have been playing an old rpg system called The Fantasy Trip. It uses a 3d5 system for its attacks. In order to determine if you hit a target enemy you roll a 3d6 which must end up being less than your own dexterity. A players dexterity is typically between 8 and 14. Typically, the lower the dexterity the higher the strength of the character. Furthermore, if you roll a 3 than it is an automatic hit and triple damage. If you roll a 4 than it is double damage. On the other end, if you roll an 18 you break your weapon. If you roll a 17, you drop your weapon. After you roll to see if you hit, you then roll certain die for certain weapons to determine damage. I love the added element of the bell curve.

Anonymous said...

TYPO-O ABOVE. I mean 3d6 not 3d5

Anonymous said...

[Warning: Purists will probably pooh-pooh this entire post. Too bad. End Disclaimer.]

A system we used to use for ability scores was 4d6 and drop the lowest one. This yields slightly higher results (the theory being that you're roleplaying a combat-capable hero, not an "average joe" waitress, farmer, or librarian.)

Here are the probabilities for outcomes of this system:

4d6 drop the lowest
3 0.077%
4 0.309%
5 0.772%
6 1.620%
7 2.932%
8 4.784%
9 7.022%
10 9.414%
11 11.420%
12 12.886%
13 13.272%
14 12.346%
15 10.108%
16 7.253%
17 4.167%
18 1.620%

3-8 10.494%
9-14 59.336%
15-18 23.148%
Avg result 12.24

For kicks, I also ran the numbers on probability for 5d6 drop the lowest two, so that I could compare them.

5d6 drop the lowest two
3 0.013%
4 0.064%
5 0.193%
6 0.527%
7 1.157%
8 2.186%
9 3.807%
10 6.044%
11 8.552%
12 11.330%
13 13.567%
14 14.853%
15 14.288%
16 12.024%
17 7.845%
18 3.549%

3-8 4.141%
9-14 58.153%
15-18 37.706%
Avg Result 13.43

Interesting. I haven't actually test-rolled some real-life examples of this yet because I'm at work. It's definitely shaped differently than a bell curve, though. It looks more like an ocean wave about to break. :P

Sham aka Dave said...

Fantastic work! Thanks, I might add these numbers in a future post (when I have the urge to post again).

The roll 4d6, drop the lowest thing was how we always did it back in my formative, 1st Edition days.

Gygax gave a number of alternative methods, iirc, in the 1e DMG.

I enjoy the OD&D method for reasons I've shared in other posts...but I used the 4d6 one for decades to great result. It's all relative in the end.

As quoted earlier:

The dice are your tools. Learn to use them properly, and they will serve you well.

And your numbers crunching certainly gives a better understanding.

Anonymous said...

I play battletech, which uses 2d6 for everything. In this game, there are modifiers for your movement, your enemy's movement, range, terrain, et c. and the base-to-hit of 4+, a 92% success rate, easily becomes a 10+ hit number, with a 17% chance of success. Using 2 or 3 d6, attack and defense bonuses must be closely matched to avoid the tails of the curve, or you get useless recruits that fail 90% of the time and unstoppable veterans that can hardly miss. I think d20 results in a better game.

Anonymous said...

There is a finnish roleplaying game (pen & paper) using multiple 6D system, in which 3d6 is the most used. It's called praedor. It's awesome.

Ynas Midgard said...

This is a good post; however, there is a minor miscalculation: based on your percentiles, getting an ability within 8-13 range has a probability of 67.6%, not 76.6% or what you wrote.

Sham aka Dave said...

Good catch! I must've simply added wrong. I need to edit and correct those numbers. Thanks.

Anonymous said...

Just wanted to tell you this post has been a great help to me! I know it's a very old post, but the same rules still apply and I am glad you showed the probabilities AND your thoughts on them. Helped me in some design issues I had.

So thank you!


Sham aka Dave said...

Your bump of this old post made me realize how quickly the last four years have passed. I am glad you found it useful. I enjoyed reading all the old comments again.

Anonymous said...

Could not agree more currently converting D20 to 3d6. Its going to be tricky; but I really feel it is worth it.

Anonymous said...

The Hero System (commonly associated with superhero role playing but I use it exclusively for fantasy) uses 3d6 for to-hit and for Skill Rolls and for pretty much all "task resolution" rolls.

Rolling 3d6 isn't harder than rolling 1d20 and some players actually like rolling more dice.

Lucius Alexander

House of the Palindromedary

Unknown said...

Just discovered this post while looking for percentages for 3d6 rolls while converting the d20 system of difficulty classes to GURPS, and I'm surprised to see no mention of GURPS. It's pretty much the quintessential 3d6 based system - you roll 3d6 for basically everything.