Yet another dice probability thread

[Morrus]

Sorry guys; I know I keep asking for help with this stuff. I want to get it right!

So I'm writing a page in my RPG which delves a bit into the maths. Only slightly, and many readers will skip it, but for those interested it can be an interesting insight, I think. So I figured why not include it?

So here's what I have so far. It's a very basic look at basic dice d6 dice pools. But I want to be sure I have it all correct. As you can see, I've also been unable to incorporate the exploding dice into the maths, meaning that the last two tables, especially, are showing figures which are too low.

This short section of the rulebook contains a little math and probability theory. You can skip it, though reading it may prove useful when it comes to assigning difficulty values.


A look at the statistics will assist the GM in assigning appropriate difficulty values. Consider the following facts:

• The average human attribute (without skill bonuses) is 4.
• This means that an average human adult will tend to roll between 4-24 on 4d6, with the most common result being 14.
• Because 4d6 is a bell-curve, the results will tend towards the average, and half of the time the average human will roll between 12 and 16.
• The exploding dice will tend to increase these odds, putting the 'average' roll at 16 on 4d6, rather than 14 with an average range of 14-18.

So if you want a difficulty value that the average person would succeed at half the time (perhaps breaking down a regular door), you should set it at 16. Some significant “landmarks” are as follows:

4	Lowest possible roll	Trivially easy; success guaranteed
12	Low end of most rolls
16	Average roll	Routine; about 50% success rate
18	High end of most rolls
24	Highest possible roll	Incredibly difficult; success unlikely

The same average difficulty for someone with basic training (taken the skill once, gaining +1d6) is 19, and for someone well-trained (taken it twice) is 23, and for someone specialised (gaining 3d6 or more on the check) is 27 or higher. Setting the value at higher than 24 means that the average person with no additional training or bonuses from elswhere will be very unlikely to succeed (barring a lucky string of exploding dice), but someone with a couple of ranks in that skill has a reasonable chance of doing so.

Average untrained person succeeds half the time, trained person succeeds most times	16
Average untrained person likely fails, but someone with basic training succeeds half the time	19
Average untrained person likely fails, but well-trained person succeeds half the time	23
Trained person likely fails, but specialized person succeeds half the time	27

Here are the average die rolls for varying sized dice pools. This is a good guide as to what is a difficult or easy target value for a given sized dice pool.

1d6	2d6	3d6	4d6	5d6	6d6	7d6	8d6	9d6	10d6
3.5	7	10.5	14	17.5	21	24.5	28	31.5	35

Finally, here's a look at the performance of an average, untrained human (i.e. one rolling 4d6) and his likelihood of hitting certain difficulty values.

4	5	6	7	8	9	10	11	12	13	14
100%	99.9%	99.6%	98.4%	97.3%	94.6%	90.2%	84.1%	76.1%	66.4%	55.6%
15	16	17	18	19	20	21	22	23	24
44.4%	33.6%	23.9%	15.9%	9.7%	5.4%	2.7%	1.2%	0.4%	0.1%

Exploding dice add about 15% (on average) to those chances, although less so at the higher extremes.

 

[Storminator]

With exploding d6, each d6 has an average of 4.2 instead of 3.5, so that fixes one of the tables. I'd have to think about the distribution a bit to get it right.

PS

 

[Umbran]

Dude, black text on a black background!

 

[Nagol]

Exploding dice also "stretch out" the curve. You can see in on anydice.com. About half the results fall between 12-18 with 4 dice exploding.

Use "output 4d[explode d6]" to see the 4 dice exploding and compare to "output 4d6" for the regular curve.

One thing I think is interesting, the most common roll (mode) for 4 exploding dice is less than the most common roll for 4d6 -- 13 vs. 14 even though the average increases.