Sunday, August 21, 2011

d20 vs. 3d6 Attribute Checks

Hey math-heads! Maybe you can help me figure out what the difference is between attribute checks rolled using 1d20 as opposed to those rolled with 3d6. I can figure out the first part since the d20 provides a linear probability curve.*

For example, if I have a 14 strength, pretty good, and have to roll 14 or under on a d20, that's 14 chances in 20, or a 70% chance of success.

If I am a weak little fellow with a STR of 6, my 6 chances in 20 = 30% success rate.

STR 11 = 55% success rate on a d20 attribute check, and STR 18 = 90% success rate on a d20 attribute check.

Got it.  Pretty reasonable-seeming mechanic.

But while I understand the concept of the bell curve and know that the results of several 3d6 rolls average out toward the middle of the range over successive rolls, I cannot quite conceive how that impacts the chances for a STR 18 attribute check vs. a STR 11 attribute check. Here is Gygax's table:

Am I correct to interpret this as meaning that a STR 11 fighter has a 13% success rate on a 3d6 attribute check, and that a STR 18 fighter has a 1% success rate?  I love d6-based mechanics like 2d6 Thievery and 2d6 Bardery but this 3d6 vs. d20 attribute check thing seems kind of extreme -- like a 3d6 attribute check mechanic would be way unfair to characters with attributes in the 12-18 range.  


* I'm no mathematician -- I stole everything I know about how dice generate probabilities from EGG's discussion of the topic on pp. 9-10 of the Dungeon Master's Guide.


  1. Let the computers do the work for you!

    Gygax's table tells you the rough percentage chance of rolling the exact number on 3d6, but what you really want to be doing is adding all the lower numbers in too. A strength 11 fighter is is going to have a 3d6 success rate of about 63%.

  2. That chart is the chance of just rolling equal to that number.

    On 3d6, the most rolled score is 10-11, as the average is 10.5, try this site out:

  3. No, that's the chance to roll that number. So you'd have a less than 1% chance to roll an 18 on any given 3d6 throw.

    If you're doing a 3d6 roll under an ability score thing, an 18 would technically never fail, although most folks using such a rule would likely institute an "18 always fails thing."

    There are 216 possible combinations of numbers (permutations) for a roll of 3d6. Only one of those gives an 18 (6,6,6) so it's actually slightly less than 1/2 a percent chance to roll an 18, or a 3 (1,1,1). There are lots of combinations that result in a 10 or 11, so the percentages of rolling one of them are much higher.

    So, for example, there are 215 combinations that result in a 17 or lower, so you'd have about a 99.5% chance of success with a score of 17.

    I'm not really a math geek, I was just good at it in school, so someone else will likely explain it better than me. But hopefully this will help.

  4. Gazumped by Kelvin and Sean. That'll teach me to write a long reply and check my math on the calculator!

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  6. No, your explanation is a good one, whereas I took the easy route of providing a link to a computer brain!

  7. THANKS for the help, mathematicians! I now see how simple it really is, but my brain usually needs assistance to overcome mathematical obstacles, no matter how simple. Thanks also for the links, Kelvin and Sean.

  8. As an aside, I've been toying with 4d6, 5d6 and 6d6 rolls under attributes to simulate more difficult obstacles. I haven't quite codified it yet, but I do think that the 3d6 idea is better than d20 any day.

  9. How about:

    2d6 for difficult tasks if you have the particularly skilled/experienced in doing that kinda thing.

    3d6 for average difficulty

    4d6 for really hard

  10. I find this site very helpful

  11. This is the short version: 3d6 attribute checks make high attribute scores even better and low ones even worse.

    It also has a range issue: if you're trying to equal to or under your ability score, then people with 18 never fail. If you're trying to roll under, people with 3 never succeed.

    Assuming you're trying to equal to or under your score, and using 17 as the high stat to avoid the range issue:

    With d20, a terrible person (stat 3) still succeeds 15% of the time (3 time in 20), an average persion (rounded to 11) succeeds 55% of the time (11 times in 20) and a great person (stat 17) succeeds 85% of the time (17 times in 20).

    With 3d6, a terrible persion (stat 3 only succeeds 0.04% of the time (1 time in 216), an average person succeeds 62.5% of the time (135 times in 216) and a great person (stat 17) succeeds about 99.53% of the time (215 times in 216).

    With 3d6, your stat needs to be at least 7 before you have a 10% chance of success, and if needs to be under 14 before you have a 10% chance of failure.

    And remember, the guy with 18 can't fail at all.

    So, basically: don't do this unless you want to make attributes extremely important. Characters with stats of 14 and up will become gods among men, night immune to failure, and those characters cursed with a stat of 7 or below can be immediately consigned to lives of dismal failure.

  12. "And remember, the guy with 18 can't fail at all."

    Exactly. That's why d20 is really used. A 20 is an auto failure. Just reverse the numbers. :)

  13. Wow, thanks for the further comments, all very enlightening. Jim, I look forward to hearing more about your 3d6, 4d6, 4d6 system if you ever hone it down.

  14. Carter, the Secret Fire rpg hat just came out uses the variable d6 system and roll under your statfor success, based on task difficulty. I find it very interesting and it gives greater meaning between an 11 or a 12 in stat for example. It also makes 3d6 straight characters more viable and interesting. It also opens up an avenue to try moves (whether combat or action task) of increased difficulty by voluntarily adding more d6s to your attempt, for example a cleave or a trick bow shot, or something like swinging down from a chandelier to attack. It's just different than adding or subtracting modifiers to a d20 roll.

  15. I agree with bombshelter13: don't do this unless attributes should be super important. Another way to think of it: if you want the bell curve to match expected results, then you can't roll attributes using 3d6 and test them using 3d6—one of the two needs to haves a linear distribution like a d20.

  16. Two points:

    1. The variable d6 idea was used in The Fantasy Trip.
    2. If you'd like a curve instead of linear so idea are 2d10 or 4d6-4 (resulting in 0-20) although the math on the later get ugly. You might find 0-5 dice. Another possibility is 4d5 (

  17. Thanks for kicking this off, Carter. The comments have been most helpful to my pondering, so thanks to everyone.