On the newly formed Praxis forum, user Keven Allen Jr asked:

I want a chart of every possible three die outcome of rolling 3d6 (order doesn’t matter)…. My mind doesn’t really love looking at numbers and percentages, but is very good at processing visual information….

As the notion of presenting probability information for role-playing games in a visual way dovetails with a possible future DivNull project, we dusted off the (rusty) graphic design tools and went to work (click for a larger version):

Direct download links:

- Large PNG (184 KB)
- Compressed SVG (11 KB)
- Adobe Illustrator source (407 KB)

why are ‘duplicates’ shown? in the context of RPGs (or really any context i can think of) you don’t care which die rolled what. you only care about the total of the dice.

as such, here is the revised table in text form:

18 = (6,6,6)

17 = (6,6,5)

16 = (6,6,4),(6,5,5)

15 = (6,6,3),(6,5,4),(5,5,5)

14 = (6,6,2),(6,5,3),(6,4,4),(5,5,4)

13 = (6,6,1),(6,5,2),(6,4,3),(5,5,3),(5,4,4)

12 = (6,5,1),(6,4,2),(6,3,3),(5,5,2),(5,4,3)

11 = (6,4,1),(6,3,2),(5,5,1),(5,4,2),(5,3,3),(4,4,3)

10 = (6,3,1),(6,2,2),(5,4,1),(5,3,2),(4,4.2),(4,3,3)

09 = (6,2,1),(5,3,1),(5,2,2),(4,4,1),(4,3,2)

08 = (6,1,1),(5,2,1),(4,3,1),(4,2,2),(3,3,2)

07 = (5,1,1),(4,2,1),(3,3,1),(3,2,2)

06 = (4,1,1),(3,2,1),(2,2,2)

05 = (3,1,1),(2,2,1)

04 = (2,1,1)

03 = (1,1,1)

*also note that 12 and 9 in the graphic chart have entries that are absolutely incorrect:

12 = (4,3,6) and 9 = (4,1,3)

i wasn’t really looking for those inaccuracies. there could be others.

Duplicates are shown because they matter to the probability a lot. The “table” as you call it isn’t a table of results: it is a bar graph. The length of the bar indicates how many combinations produce that result (i.e. the probability of that result). Your revised table doesn’t do that at all.

You are correct about the mistakes. The entry in 12 should, of course, be 4,3,5 and the one for 9 should be 4,1,4. I’ll correct this when I get a moment.

table/bar graph, tom-A-to/tom-ah-to, whatever…

the duplicates DO NOT matter if you are throwing 3 die at one time. if i roll a 6, 5, and 4, it does not matter which die was the 6 or which die was the 5 or which die was the 4.

…and yes, my table DOES indicate the probability of result just as much as the length of a bar. just look at the length of the line. i’m sorry if you need a “pretty picture” to see it.

The duplicates ABSOLUTELY DO matter to the

probabilityof the outcomes, even if they don’t matter to theresult. For example, say your 3d6 are different colors, one red die, one blue die, one green die. Lets say you roll these and they add up to 17. You are right that any game you play wont really care which color die came up which; only the resulting sum matters to the outcome in the game. However, this chart isn’t representing possible outcomes. It is representing theprobabilityof those outcomes. And, to calculate that, it really does matter which color comes up which.Take a look at your table. It lists one result summing to 18 (i.e. all dice come up six), and one result summing to 17 (i.e. two dice come up six, one comes up five). Does this mean that rolling a 17 and an 18 are equally likely? Of course not.

You can only get an 18 one way: the red die is six, the blue die is six and the green die is six. You can list this result as a (red, green, blue) triplet: (6,6,6).

On the other hand you can get a 17

three distinctways: (6,6,5), (5,6,6) or (6,5,6). In other words, you are three times more likely to get a 17 than an 18 on a 3d6. To figure out the probability, you have to count the distinct combinations, not just the results.Still don’t believe me? Then lets play this game. We start rolling 3d6 over and over. Any time a 17 comes up, you pay me a dollar. Any time an 18 comes up, I pay you

two dollars. If your result chart really is an accurate representation of the probability, then you can’t lose. Since your chart says that 18 and 17 are equally likely, you should earn twice as much money as you pay, assuming we track a decent sample of rolls (say 1000 of them).I, on the other hand, will play this game with you any time you like, because I know that my probability chart correctly illustrates the chances of rolling a number on a 3d6. Since 17 will happen three times more often than 18, for every two dollars you earn, you will pay me three, on average.

You can verify this by just rolling the dice 100 or 1000 times (or having a computer do it for you) and track the results.

It might be over a year late, but I just found it now (while Googling “3d6 probabilities” for a visual aid to help me in a Statistics class), so thank you for the very nice chart, Wordman! That’s pretty much exactly what I was hoping for!

And also, thank you very much for correctly explaining why the different combinations DO matter — because you wouldn’t believe how often I have had the very same argument, and that “duplicates don’t matter” nonsense must be squashed at every available opportunity. I cannot fathom why people like Adam don’t understand such an incredibly simple point, and my only guess is that they’ve never actually taken any Math classes that dealt with Probability. Or even bothered to THINK about it for a minute.

Wordman’s graph can also be used to determine probability to show how a +1 or -1 can make such a difference with the roll. There are 216 combinations of rolls that can be done with three d6’s (6x6x6). Because there is only one way to roll a 3 and one way to roll an 18, there is a 100% of rolling 3 or higher, or 18 or lower.

Let’s say you are playing a system where you want your roll to be lower than your ability, like in the Hero system. A plus (+) modifier will adjust what you have to roll or less, thereby increasing your chances. So, say a player wants his character to do something and he has to get a 12 or less. Using Wordman’s chart, we find there are 160 rolls that will yield these numbers. 160/216 is a 74% chance.

In Adam’s chart, there are 54 possible outcomes, 38 of which total 12 or less. 38/54 is 70%. In my book, 74 does not equal 70.

When it comes to distribution of dice totals and your character’s chances of success or doing more damage, you have to reference Wordman’s chart, which is a permutation. That chart may be missing percentages, but it really does chance the percentage chance of getting X or lower, or X or higher.

I looked all over the internet for a probability chart of rolling “1”‘s on a 3d6, out of all 216 results, for very specific wargaming reasons (axis&allies). So correct me if i’m wrong: there are 77 ways of rolling at least one “1” (~35%), 14 ways to get two 1’s (~6%) and one way to get all three (0.46%)? I’m terrible at math, so i counted them individually on your chart lol.

Also, Adam’s comments are hilarious. I lost count of how many times i had this discussion.

there is a 42.1% chance of at least 1 one showing: 1-(5/6*5/6*5/6)

the odds of getting 2 ones: 1-(5/6) * 1-(5/6*1-5/6) = 0.1667*0.306 = ~5%

odds of getting 3 ones: 1/6 * 1/6 * 1/6 = 0.0046 or about 0.46%

Hey, I’m resurrecting this for a really selfish reason.

Does anyone have the original data, I really don’t feel like typing it all up manually. ðŸ™‚

I’m needing both the unique results (all 216 combinations) AND the die-generic results (where it doesn’t matter which die has what number).

I’m working on a dice variant where both the individual results (if they have double, triple, or straights) and the sum matter.

I’ve put it all in excel, I have a giant chart, but I’m having trouble parsing it so I can remove duplicates…