# Roll 3

As a follow up to a long ago post about Dice probabilities, we derive 6^{n} as a formula to use to calculate the possible outcomes.

Two dice 6^{2} = 36 outcomes.

Three dice 6^{3} = 216 outcomes.

If we take 3 Dice, the results would range from 3 – 18, *or between (n* – 6^{n}).

The tricky part comes from trying to calculate the chances of each result without doing the mathematically inept person I am did, which was to list out all the possibilities.

- 3 = 1 + 1 + 1
- 4 = 1 + 1 + 2
- 5 = 1 + 1 + 3 = 2 + 2 + 1
- 6 = 1 + 1 + 4 = 1 + 2 + 3 = 2 + 2 + 2
- 7 = 1 + 1 + 5 = 2 + 2 + 3 = 3 + 3 + 1 = 1 + 2 + 4
- 8 = 1 + 1 + 6 = 2 + 3 + 3 = 4 + 3 + 1 = 1 + 2 + 5 = 2 + 2 + 4
- 9 = 6 + 2 + 1 = 4 + 3 + 2 = 3 + 3 + 3 = 2 + 2 + 5 = 1 + 3 + 5 = 1 + 4 + 4
- 10 = 6 + 3 + 1 = 6 + 2 + 2 = 5 + 3 + 2 = 4 + 4 + 2 = 4 + 3 + 3 = 1 + 4 + 5
- 11 = 6 + 4 + 1 = 1 + 5 + 5 = 5 + 4 + 2 = 3 + 3 + 5 = 4 + 3 + 4 = 6 + 3 + 2
- 12 = 6 + 5 + 1 = 4 + 3 + 5 = 4 + 4 + 4 = 5 + 2 + 5 = 6 + 4 + 2 = 6 + 3 + 3
- 13 = 6 + 6 + 1 = 5 + 4 + 4 = 3 + 4 + 6 = 6 + 5 + 2 = 5 + 5 + 3
- 14 = 6 + 6 + 2 = 5 + 5 + 4 = 4 + 4 + 6 = 6 + 5 + 3
- 15 = 6 + 6 + 3 = 6 + 5 + 4 = 5 + 5 + 5
- 16 = 6 + 6 + 4 = 5 + 5 + 6
- 17 = 6 + 6 + 5
- 18 = 6 + 6 + 6

So the grand result would be;

- Probability of a sum of 3: 1/216 = 0.5%
- Probability of a sum of 4: 3/216 = 1.4%
- Probability of a sum of 5: 6/216 = 2.8%
- Probability of a sum of 6: 10/216 = 4.6%
- Probability of a sum of 7: 15/216 = 7.0%
- Probability of a sum of 8: 21/216 = 9.7%
- Probability of a sum of 9: 25/216 = 11.6%
- Probability of a sum of 10: 27/216 = 12.5%
- Probability of a sum of 11: 27/216 = 12.5%
- Probability of a sum of 12: 25/216 = 11.6%
- Probability of a sum of 13: 21/216 = 9.7%
- Probability of a sum of 14: 15/216 = 7.0%
- Probability of a sum of 15: 10/216 = 4.6%
- Probability of a sum of 16: 6/216 = 2.8%
- Probability of a sum of 17: 3/216 = 1.4%
- Probability of a sum of 18: 1/216 = 0.5%