About Rolling Two Dice
Rolling two dice is one of the most fundamental probability experiments and is central to countless board games including Monopoly, Craps, Settlers of Catan, and Backgammon. When two six-sided dice are rolled, there are 36 possible equally likely outcomes (6 x 6), producing sums ranging from 2 to 12.
The distribution of sums is not uniform. The sum of 7 is the most probable outcome, occurring in 6 out of 36 combinations (16.67%). The sums of 2 and 12 are the least probable, each occurring only once (2.78%).
Probability Formulas for Two Dice
For two standard six-sided dice, the expected sum is 7. The variance of the sum is 2 x (35/12) = 5.833, and the standard deviation is approximately 2.415.
Sum Probability Table (Two 6-Sided Dice)
| Sum | Combinations | Probability |
|---|---|---|
| 2 | 1 | 2.78% |
| 3 | 2 | 5.56% |
| 4 | 3 | 8.33% |
| 5 | 4 | 11.11% |
| 6 | 5 | 13.89% |
| 7 | 6 | 16.67% |
| 8 | 5 | 13.89% |
| 9 | 4 | 11.11% |
| 10 | 3 | 8.33% |
| 11 | 2 | 5.56% |
| 12 | 1 | 2.78% |
Applications
- Board Games: Monopoly, Risk, Settlers of Catan, and Backgammon all rely on two-dice rolls.
- Craps: The casino game is built entirely around the probability distribution of two dice.
- Probability Education: Two-dice experiments are the classic introduction to discrete probability distributions.
- Decision Making: Random two-dice rolls can be used for fair randomized selection processes.
Frequently Asked Questions
What is the most common sum when rolling two dice?
The most common sum is 7, which can be formed by 6 different combinations: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). This gives a probability of 6/36 = 16.67%.
What is the probability of rolling doubles?
With two six-sided dice, there are 6 doubles possible (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) out of 36 total outcomes. The probability is 6/36 = 1/6 = 16.67%.
What is the probability of rolling snake eyes (both ones)?
Snake eyes (1,1) occurs once out of 36 possible outcomes, giving a probability of 1/36 = 2.78%.