Let’s explore how to calculate the probability of a particular dice role. When you roll a single six-sided die (also known as a d6), each side (1, 2, 3, 4, 5, or 6) has an equal chance of landing face-up, if the die is fair. A fair die is one that is unbiased. The probability of rolling any specific number is 1 out of 6, or approximately 0.167 (16.7%).
When you roll two dice, things get more interesting: There are now 36 possible outcomes. In other words, six possibilities for the first die times six possibilities for the second die.
If you want to find the probability of rolling an 8, you must find all the combinations that add up to 8. In this case, a pair of rolled dice can sum to 8 in with the following combinations: (2,6), (3,5), (4,4), (5,3), and (6,2). These 5 combinations sum to 8, so the probability is 5 out of 36, or approximately 0.139 (13.9%).
If you want both dice to land on 4, there’s only one combination that works: (4,4). The probability of this specific event is 1 out of 36, or approximately 0.028 (2.8%). Similarly, if you want to roll a 12, there’s only one combination that works: (6,6). So the probability of rolling a 12 is also 2.8%.
For the case of two, six-side dice, you can refer to the following probability table for possible outcomes:
| Sum of dice | Combinations | Probability |
|---|---|---|
| 2 | (1,1) | 1/36 |
| 3 | (1,2),(2,1) | 2/36 |
| 4 | (1,3),(2,2),(3,1) | 3/36 |
| 5 | (1,4),(2,3),(3,2),(4,1) | 4/36 |
| 6 | (1,5),(2,4),(3,3),(4,2),(5,1) | 5/36 |
| 7 | (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) | 6/36 |
| 8 | (2,6),(3,5),(4,4),(5,3),(6,2) | 5/36 |
| 9 | (3,6),(4,5),(5,4),(6,3) | 4/36 |
| 10 | (4,6),(5,5),(6,4) | 3/36 |
| 11 | (5,6),(6,5) | 2/36 |
| 12 | (6,6) | 1/36 |
Each row represents a possible sum of the numbers on the two dice, the combinations that result in that sum, and the probability of that sum occurring. The probabilities are calculated as the number of combinations that result in the sum divided by the total number of possible outcomes. For two six-sided dice, there are 36 possible outcomes.
By referring to the table, the probability of rolling a sum of 8 is 5/36 because there are 5 combinations that give a sum of 8: (2,6), (3,5), (4,4), (5,3), and (6,2).
