So I wanted to figure out the odds on a roll of 1d6d6 - that is, roll a six-sided dice, then whatever the result is, roll that many six-sided dice and add up the result. It’s a nice quick and dirty way to produce a Poisson-like low-biased probability distribution using only standard dice. However, there’s absolutely no information on Google for this, or for dice odds past 4d6, so I made a quick program to calculate it. I figured I’d stick my results up for the next person who comes looking, to save them the trouble.
Here’s the probability distribution in graphical form:
For those who prefer hard numbers, here are the odds of the given result with the given roll:
1d6d6: (279,936 total - i.e., to find the probability of a result, divide its odds by 279,936)
1 - 7,776
2 - 9,072
3 - 10,584
4 - 12,348
5 - 14,406
6 - 16,807
7 - 11,832
8 - 12,507
9 - 13,076
10 - 13,482
11 - 13,650
12 - 13,482
13 - 12,852
14 - 12,897
15 - 12,772
16 - 12,453
17 - 11,928
18 - 11,207
19 - 10,332
20 - 9,387
21 - 8,292
22 - 7,101
23 - 5,880
24 - 4,697
25 - 3,612
26 - 2,667
27 - 1,876
28 - 1,251
29 - 786
30 - 462
31 - 252
32 - 126
33 - 56
34 - 21
35 - 6
36 - 1
And here’s the breakdowns of the individual rolls.
1d6: (6 total)
1 - 1
2 - 1
3 - 1
4 - 1
5 - 1
6 - 1
2d6: (36 total)
2 - 1
3 - 2
4 - 3
5 - 4
6 - 5
7 - 6
8 - 5
9 - 4
10 - 3
11 - 2
12 - 1
3d6: (216 total)
3 - 1
4 - 3
5 - 6
6 - 10
7 - 15
8 - 21
9 - 25
10 - 27
11 - 27
12 - 25
13 - 21
14 - 15
15 - 10
16 - 6
17 - 3
18 - 1
4d6: (1296 total)
4 - 1
5 - 4
6 - 10
7 - 20
8 - 35
9 - 56
10 - 80
11 - 104
12 - 125
13 - 140
14 - 146
15 - 140
16 - 125
17 - 104
18 - 80
19 - 56
20 - 35
21 - 20
22 - 10
23 - 4
24 - 1
5d6: (7,776 total)
5 - 1
6 - 5
7 - 15
8 - 35
9 - 70
10 - 126
11 - 205
12 - 305
13 - 420
14 - 540
15 - 651
16 - 735
17 - 780
18 - 780
19 - 735
20 - 651
21 - 540
22 - 420
23 - 305
24 - 205
25 - 126
26 - 70
27 - 35
28 - 15
29 - 5
30 - 1
6d6: (46,656 total)
6 - 1
7 - 6
8 - 21
9 - 56
10 - 126
11 - 252
12 - 456
13 - 756
14 - 1161
15 - 1666
16 - 2247
17 - 2856
18 - 3431
19 - 3906
20 - 4221
21 - 4332
22 - 4221
23 - 3906
24 - 3431
25 - 2856
26 - 2247
27 - 1666
28 - 1161
29 - 756
30 - 456
31 - 252
32 - 126
33 - 56
34 - 21
35 - 6
36 - 1
For those looking to do similar things at home, the quick way is to use a site like this one that will generate a probability table, and the more generic way is to write a program to do it yourself. I used C# as my language, and the 6d6 loop is as follows:
public static Random die = new Random();
static void Main()
{
int[] results = { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 };
for (int a=1; a <= 6; a++)
{
for (int b = 1; b <= 6; b++)
{
for (int c = 1; c <= 6; c++)
{
for (int d = 1; d <= 6; d++)
{
for (int e = 1; e <= 6; e++)
{
for (int f = 1; f <= 6; f++)
{
results[a + b + c + d + e + f]++;
}
}
}
}
}
}
for (int n=1; n<=36;n++)
{
Console.WriteLine(results[n]);
}
Console.ReadLine();
}
I used Excel to put all the various combinations together to produce the chart above.
