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What Are The Odds When Rolling 3d6?

posted Tuesday, May 25th 2010 by Jonathan Drain
None of the Above

You kids today have it easy. Back in the early days of D&D you rolled 3d6 for ability scores, placed in the order you rolled them. No “4d6 drop lowest” or “arrange as desired” – if you roll a 3 for Strength, your character has 3 Strength. Looks like you aren’t playing a fighter this time.

Labyrinth Lord (free download here) is a Basic D&D retro-clone that still uses 3d6 for ability score generation. By third edition or fourth edition D&D standards, LL is quite deadly: poison kills you outright, and dragon’s breath weapons deal their current hit points in damage.

But what are the odds of rolling each ability score on 3d6? I wrote up a quick python script like my 4d6 drop lowest calculator from 2006.

Firstly, the average roll on 3d6 is 10.5, compared to 12.244 on 4d6 drop lowest. The “human average” +0 modifier in third edition D&D comes from this result.

The odds of rolling an 18 are an unlikely 0.463%, or one in 216. The odds of rolling at least one 18 are 2.746%. In both cases this is about 3.5 times harder than third edition D&D’s 4d6 drop lowest. It’s around 20 times harder to get an 18 in a particular ability score, since early D&D required that you assign the ability scores in the order they’re rolled.

Here’s the full chart of results of 3d6:

Score   Freq   Percentage
-----   ----   ----------
3          1       0.463%
4          3       1.389%
5          6       2.778%
6         10       4.630%
7         15       6.944%
8         21       9.722%
9         25      11.574%
10        27      12.500%
11        27      12.500%
12        25      11.574%
13        21       9.722%
14        15       6.944%
15        10       4.630%
16         6       2.778%
17         3       1.389%
18         1       0.463%

Comments

  1. MJ Harnish

    May 25th, 2010

    One minor correction: 10.5 is not the average roll on 3d6. It’s the mean of the most frequent (common) outcomes on the probability distribution (10 & 11 being the more likely results when you roll 3d6).

    The most interesting difference between 3d6 and 4d6 IMO, is that 3D6 generates a normal distribution while 4D6, discard the lowest, creates a negatively skewed distribution which means the probability of any outcome leans towards better scores creating more “heroic” characters.

    It’s important though to realize that the distribution and significance of modifiers associated with ability scores is quite different between 3rd edition & older versions of D&D. In 3rd (and later) edition, modifiers associated with high ability scores are much more important because they’re factored in to the CRs of encounters in a formulaic way. In earlier editions there’s almost no mathematical consideration of game balance (instead it was left up to DM’s discretion) and so high ability scores aren’t nearly as critical (though they certainly help!).

  2. Michelle

    May 25th, 2010

    High ability scores didn’t buy you much in D&D either. Wizards found it easier to learn spells, and you could get a small XP bonus if you had a high score in your main ability.

    D&D was an odd game, for sure. I remember in one of our first sessions, one of my brothers rolled up a pair of halflings named Honest Bob and Friendly Bob (think used-car sales) who each had exactly 1 hit point. I don’t remember the names of his next characters, but he rolled them up about 15 minutes later, as the Bobs had been trundled off to Lemon Law Hell.

  3. When Do Smarter Wizards Deserve More XP? « Jonathan Drain’s D20 Source: Dungeons & Dragons Blog

    May 29th, 2010

    [...] I said previously on the odds of each result when rolling 3d6, D&D was a lot more lethal back in 1981. Your character can start with as little as one hit [...]

  4. gurps lover

    November 29th, 2010

    Just play gurps. Build your character and hope that you don’t have a killer gm (meaning he’s out to destroy your characters).

  5. Jackalope

    May 18th, 2013

    @MJ Harnish
    The term “average” in mathematics can describe mean, median, or mode. The average IS 10.5 when calculated as a mean. The average roll is 10 or 11 when calculated as a median or mode.

    10.5 is arguably the more useful way to think of it, because the easiest way to think of d6 is having 3.5 as the average roll. Average roll of 5d6? 3.5 x 5 = 17.5

  6. Kurt

    January 4th, 2014

    I hate to be “that guy”, but rolling 3d6 only is a myth with 1E (unless your talking original White box…). On page 11 of the DMG there’s 4 different ‘official’ and recommended methods for rolling up characters that give you better odds than 3d6. Method 1 is 4d6 drop the lowest.

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