5d6 Bell Curve

[Quartz]

I don't quite understand what I'm doing there. The middle values occur 10% of the time.

True, but you're not trying to get exactly the mean value, are you? You're trying to get the mean value or more. That means that you have to exclude everything that's less than the mean value. It's actually easier with a smooth curve: pull up a graph of a normal distribution and place a piece of paper over half of it (i.e. the 18 point) and look at the area under the curve and compare it with the area under the curve that's hidden by the paper. Now move the paper along the X axis to get a feel for how the areas change.

 

[Morrus]

I think I've got a handle on what I want to do.

So, putting this together, this seems to be a first-pass summary of the difficulty value assignment on 5d6. I'm currently situations, or non-average people who might mean rolls of more or less than 5d6.

The difficulty value of a standard task that most people would succeed at 50% of the time is 18. Breaking a door down, climbing a tree, jumping the gap between two buildings, attempting to navigate official bureaucracy, or getting an excited dog to sit.

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

• The average human attribute (without skill bonuses) is 5.
• This means that an average human adult will tend to roll between 5-30 on 5d6, with the most common result being 18.
• Because 5d6 is a bell-curve, the results will tend towards the average, and half of the time the average human will roll between 15 and 20.

So if you want a difficulty level that the average person would succeed at half the time (perhaps breaking down a regular door), you should set it at 18. Some significant “landmarks” are as follows:


[TABLE="width: 100%"] [TR] [TD="bgcolor: #e6e6ff"]

5​

[/TD] [TD="bgcolor: #e6e6ff"]

Lowest possible roll​

[/TD] [TD="bgcolor: #e6e6ff"]

Trivially easy; success guaranteed​

[/TD] [/TR] [TR] [TD]

15​

[/TD] [TD]

Low end of most rolls​

[/TD] [TD]

​

[/TD] [/TR] [TR] [TD="bgcolor: #e6e6ff"]

18​

[/TD] [TD="bgcolor: #e6e6ff"]

Average roll​

[/TD] [TD="bgcolor: #e6e6ff"]

About 50% success rate​

[/TD] [/TR] [TR] [TD]

20
​

[/TD] [TD]

High end of most rolls​

[/TD] [TD]

​

[/TD] [/TR] [TR] [TD="bgcolor: #e6e6ff"]

30​

[/TD] [TD="bgcolor: #e6e6ff"]

Highest possible roll​

[/TD] [TD="bgcolor: #e6e6ff"]

Incredibly difficult; success unlikely​

[/TD] [/TR] [/TABLE] More difficult tasks have a higher difficulty value. However, no matter how difficult a task, the exploding dice mean that there's always a chance, even if it's very small.

[TABLE="width: 100%"] [TR] [TD="bgcolor: #99ccff"] Sample Task
[/TD] [TD="bgcolor: #99ccff"] Difficulty Value
[/TD] [/TR] [TR] [TD="bgcolor: #e6e6ff"] Picking up a heavy book
[/TD] [TD="bgcolor: #e6e6ff"] 5
[/TD] [/TR] [TR] [TD] Hearing a loud voice just outside
[/TD] [TD] 10
[/TD] [/TR] [TR] [TD="bgcolor: #e6e6ff"] Breaking down a standard pine door
[/TD] [TD="bgcolor: #e6e6ff"] 15
[/TD] [/TR] [TR] [TD] Building a campfire
[/TD] [TD] 18
[/TD] [/TR] [TR] [TD="bgcolor: #e6e6ff"] Spotting a hidden compartment or door
[/TD] [TD="bgcolor: #e6e6ff"] 23
[/TD] [/TR] [TR] [TD] Picking a regular lock
[/TD] [TD] 25
[/TD] [/TR] [TR] [TD="bgcolor: #e6e6ff"] Swimming a fast-flowing river
[/TD] [TD="bgcolor: #e6e6ff"] 25
[/TD] [/TR] [TR] [TD] Building a starship engine
[/TD] [TD] 30
[/TD] [/TR] [TR] [TD="bgcolor: #e6e6ff"] Hacking into government systems
[/TD] [TD="bgcolor: #e6e6ff"] 40*

[/TD] [/TR] [/TABLE] *Obviously, unless additional dice are rolled, an average attribute with no trained skill will not be able to do this (excepting luck from exploding dice).

 

[Nagol]

Just remember that the distribution means variation from the centre means a lot for the first few points and then falls rapidly.

That means in this case, a 1 point penalty applies a greater than 10% reduction in probability if it moves the success point from 18 to 19.

Here is a table of success probabilities. Note that spotting a hidden door falls to about 10% and picking a standard lock falls to about 3% barring any bonuses. You "low-end for most rolls" fails about 1 in 4 times.

Target  Chance
5	100.00%
6	99.99%
7	99.92%
8	99.73%
9	99.28%
10	98.38%
11	96.76%
12	94.12%
13	90.20%
14	84.80%
15	77.85%
16	69.48%
17	60.03%
18	50.00%
19	39.97%
20	30.52%
21	22.15%
22	15.20%
23	9.80%
24	5.88%
25	3.24%
26	1.62%
27	0.72%
28	0.27%
29	0.08%
30	0.01%

 

[Morrus]

Just remember that the distribution means variation from the centre means a lot for the first few points and then falls rapidly.

That means in this case, a 1 point penalty applies a greater than 10% reduction in probability if it moves the success point from 18 to 19.

At present, there will never be flat penalties - just adjustments to the dice pool. So you'd be adding or subtracting d6s (or about 3.5 on average). So it's a little more complex even than that!

Note that spotting a hidden door falls to about 10% and picking a standard lock falls to about 3% barring any bonuses.

Plus exploding dice, but that sounds about right. I'd guess that I (an average person with no thievery skills) would have a 3% chance or less to pick a lock. What does adding 1d6 or 2d6 to that roll do?
​

 

[Nagol]

At present, there will never be flat penalties - just adjustments to the dice pool. So you'd be adding or subtracting d6s (or about 3.5 on average). So it's a little more complex even than that!

Plus exploding dice, but that sounds about right. I'd guess that I (an average person with no thievery skills) would have a 3% chance or less to pick a lock.
​

If the dice are added/subtracted straight, the swings around the base 18 target are large.

Dropping a die to 4d6 means the character will succeed a DC18 check less than 1 in 6 times (~15.9%).
Adding a die to 6d6 means the character will succeed more than 3 in 4 attempts (~79.4%)

Note that this is without exploding dice results (as opposed to dice filled with nitroglycerin). That makes the math more messy.

 

[Morrus]

If the dice are added/subtracted straight, the swings around the base 18 target are large.

Dropping a die to 4d6 means the character will succeed a DC18 check less than 1 in 6 times (~15.9%).
Adding a die to 6d6 means the character will succeed more than 3 in 4 attempts (~79.4%)

That sounds like it's working, then. Attempting the average task while trained means you'll probably be able to do it. Attempting it while disadvantaged means you probably won't. That latter is maybe a tiny bit steeper than I'd intended, but it's not way out there (and the exploding dice will bring it up a tiny bit).

 

[Nagol]

I had a few minutes this afternoon to play around with the scenario using exploding dice.

My assumption for exploding dice is on a roll of a 6, another die was rolled and added to the total and that continued until no further sixes were rolled. I capped the maximum at 64+ points.

I ran a million result set and came up with the following probability charts:

For 5d6:

Result	Chance	This or lower
4	0.0000%	0.0000%
5	0.0143%	0.0143%
6	0.0693%	0.0836%
7	0.1983%	0.2819%
8	0.4537%	0.7356%
9	0.8963%	1.6319%
10	1.5496%	3.1815%
11	2.3896%	5.5711%
12	3.3756%	8.9467%
13	4.2853%	13.2320%
14	5.1132%	18.3452%
15	5.6468%	23.9920%
16	5.9765%	29.9685%
17	6.0853%	36.0538%
18	6.0031%	42.0569%
19	5.8706%	47.9275%
20	5.6901%	53.6176%
21	5.3724%	58.9900%
22	5.0159%	64.0059%
23	4.6288%	68.6347%
24	4.1808%	72.8155%
25	3.7215%	76.5370%
26	3.3078%	79.8448%
27	2.9394%	82.7842%
28	2.5628%	85.3470%
29	2.2617%	87.6087%
30	1.9705%	89.5792%
31	1.6802%	91.2594%
32	1.4488%	92.7082%
33	1.2145%	93.9227%
34	1.0216%	94.9443%
35	0.8762%	95.8205%
36	0.7291%	96.5496%
37	0.6004%	97.1500%
38	0.5224%	97.6724%
39	0.4152%	98.0876%
40	0.3365%	98.4241%
41	0.2911%	98.7152%
42	0.2378%	98.9530%
43	0.1982%	99.1512%
44	0.1641%	99.3153%
45	0.1288%	99.4441%
46	0.1093%	99.5534%
47	0.0834%	99.6368%
48	0.0728%	99.7096%
49	0.0560%	99.7656%
50	0.0430%	99.8086%
51	0.0357%	99.8443%
52	0.0338%	99.8781%
53	0.0235%	99.9016%
54	0.0176%	99.9192%
55	0.0167%	99.9359%
56	0.0125%	99.9484%
57	0.0101%	99.9585%
58	0.0084%	99.9669%
59	0.0071%	99.9740%
60	0.0047%	99.9787%
61	0.0040%	99.9827%
62	0.0027%	99.9854%
63	0.0035%	99.9889%
64	0.0111%	100.0000%

4d6

Result	Chance	This or lower
4	0.0822%	0.0822%
5	0.3160%	0.3982%
6	0.7751%	1.1733%
7	1.5743%	2.7476%
8	2.7164%	5.4640%
9	4.0105%	9.4745%
10	5.2899%	14.7644%
11	6.3635%	21.1279%
12	7.0969%	28.2248%
13	7.2107%	35.4355%
14	7.0459%	42.4814%
15	6.7111%	49.1925%
16	6.2232%	55.4157%
17	5.7220%	61.1377%
18	5.3536%	66.4913%
19	4.8660%	71.3573%
20	4.2929%	75.6502%
21	3.7931%	79.4433%
22	3.2663%	82.7096%
23	2.7596%	85.4692%
24	2.3864%	87.8556%
25	2.0591%	89.9147%
26	1.7645%	91.6792%
27	1.4929%	93.1721%
28	1.2207%	94.3928%
29	1.0184%	95.4112%
30	0.8515%	96.2627%
31	0.6922%	96.9549%
32	0.5767%	97.5316%
33	0.4762%	98.0078%
34	0.3858%	98.3936%
35	0.3066%	98.7002%
36	0.2495%	98.9497%
37	0.1994%	99.1491%
38	0.1684%	99.3175%
39	0.1376%	99.4551%
40	0.1112%	99.5663%
41	0.0841%	99.6504%
42	0.0689%	99.7193%
43	0.0533%	99.7726%
44	0.0484%	99.8210%
45	0.0391%	99.8601%
46	0.0295%	99.8896%
47	0.0247%	99.9143%
48	0.0173%	99.9316%
49	0.0141%	99.9457%
50	0.0114%	99.9571%
51	0.0083%	99.9654%
52	0.0073%	99.9727%
53	0.0068%	99.9795%
54	0.0037%	99.9832%
55	0.0041%	99.9873%
56	0.0031%	99.9904%
57	0.0021%	99.9925%
58	0.0022%	99.9947%
59	0.0010%	99.9957%
60	0.0010%	99.9967%
61	0.0006%	99.9973%
62	0.0009%	99.9982%
63	0.0002%	99.9984%
64	0.0016%	100.0000%

6d6

Result	Chance	This or lower
4	0.0000%	0.0000%
5	0.0000%	0.0000%
6	0.0035%	0.0035%
7	0.0129%	0.0164%
8	0.0432%	0.0596%
9	0.1218%	0.1814%
10	0.2717%	0.4531%
11	0.5390%	0.9921%
12	0.9246%	1.9167%
13	1.4511%	3.3678%
14	2.0857%	5.4535%
15	2.8246%	8.2781%
16	3.5149%	11.7930%
17	4.1072%	15.9002%
18	4.6856%	20.5858%
19	5.0541%	25.6399%
20	5.2858%	30.9257%
21	5.4137%	36.3394%
22	5.4300%	41.7694%
23	5.3415%	47.1109%
24	5.2088%	52.3197%
25	4.9789%	57.2986%
26	4.6614%	61.9600%
27	4.4041%	66.3641%
28	4.0626%	70.4267%
29	3.6902%	74.1169%
30	3.3489%	77.4658%
31	2.9935%	80.4593%
32	2.6821%	83.1414%
33	2.3611%	85.5025%
34	2.0896%	87.5921%
35	1.8154%	89.4075%
36	1.6158%	91.0233%
37	1.3908%	92.4141%
38	1.1733%	93.5874%
39	1.0354%	94.6228%
40	0.8700%	95.4928%
41	0.7449%	96.2377%
42	0.6204%	96.8581%
43	0.5189%	97.3770%
44	0.4418%	97.8188%
45	0.3733%	98.1921%
46	0.3147%	98.5068%
47	0.2627%	98.7695%
48	0.2096%	98.9791%
49	0.1881%	99.1672%
50	0.1497%	99.3169%
51	0.1259%	99.4428%
52	0.1010%	99.5438%
53	0.0857%	99.6295%
54	0.0692%	99.6987%
55	0.0528%	99.7515%
56	0.0472%	99.7987%
57	0.0390%	99.8377%
58	0.0305%	99.8682%
59	0.0239%	99.8921%
60	0.0206%	99.9127%
61	0.0140%	99.9267%
62	0.0137%	99.9404%
63	0.0124%	99.9528%
64	0.0472%	100.0000%

Some things to note:

• Normally I'd do a complete solution rather than a monte carlo, but this was faster. The results should be indicative considering the number of repetitions versus the die size.
• The cumulative chance is the chance that number or lower is rolled. To calculate the chance of rolling X or higher, find (X-1) on the chart and subtract that cumulative probability from 1.
• The exploding mechanic stretches the range of probable results somewhat.
• The exploding results shifts the 50% line for 5d6 up. 18 has a 64% success rate. 19 has a 58% success rate. 20 has a 53% success rate.
• 4d6 achieves an 18 39% of the time, a 19 33% of the time, and a 20 29% of the time.
• 6d6 hits the same targets 84%, 79%, and 74% respectively.

 

[Morrus]

Oh, that's interesting (and great - thank you!)

The exploding dice is more significant than I thought it would be. I actually like those figures a little more. Food for thought!

It's useful you've gone up to 64, too - that can help me with some of my scaling issues.