Dice provide great illustrations for concepts in probability. The most commonly used dice are cubes with six sides. Here, we will see how to calculate probabilities for rolling three standard dice. It is a relatively standard problem to calculate the probability of the sum obtained by rolling two dice. There are a total of 36 different rolls with two dice, with any sum from 2 to 12 possible. How does the problem change if we add more dice?

## Possible Outcomes and Sums

Just as one die has six outcomes and two dice have 6^{2} = 36 outcomes, the probability experiment of rolling three dice has 6^{3} = 216 outcomes. This idea generalizes further for more dice. If we roll *n* dice then there are 6^{n} outcomes.

We can also consider the possible sums from rolling several dice. The smallest possible sum occurs when all of the dice are the smallest, or one each. This gives a sum of three when we are rolling three dice. The greatest number on a die is six, which means that the greatest possible sum occurs when all three dice are sixes. The sum of this situation is 18.

When *n* dice are rolled, the least possible sum is *n* and the greatest possible sum is 6*n*.

- There is one possible way three dice can total 3
- 3 ways for 4
- 6 for 5
- 10 for 6
- 15 for 7
- 21 for 8
- 25 for 9
- 27 for 10
- 27 for 11
- 25 for 12
- 21 for 13
- 15 for 14
- 10 for 15
- 6 for 16
- 3 for 17
- 1 for 18

## Forming Sums

As discussed above, for three dice the possible sums include every number from three to 18. The probabilities can be calculated by using counting strategies and recognizing that we are looking for ways to partition a number into exactly three whole numbers. For example, the only way to obtain a sum of three is 3 = 1 + 1 + 1. Since each die is independent from the others, a sum such as four can be obtained in three different ways:

- 1 + 1 + 2
- 1 + 2 + 1
- 2 + 1 + 1

Further counting arguments can be used to find the number of ways of forming the other sums. The partitions for each sum follow:

- 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

When three different numbers form the partition, such as 7 = 1 + 2 + 4, there are 3! (3x2x1) different ways of permuting these numbers. So this would count toward three outcomes in the sample space. When two different numbers form the partition, then there are three different ways of permuting these numbers.

## Specific Probabilities

We divide the total number of ways to obtain each sum by the total number of outcomes in the sample space, or 216. The results are:

- 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%

As can be seen, the extreme values of 3 and 18 are least probable. The sums that are exactly in the middle are the most probable. This corresponds to what was observed when two dice were rolled.