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Dice odds

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:1d6d6

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.