When rolling a pair of fair dice, each die has 6 faces, resulting in a total of 36 possible outcomes (6 faces on the first die multiplied by 6 faces on the second die). The probabilities of rolling specific sums can be calculated based on these outcomes.
Here’s a breakdown of the probabilities for sums ranging from 2 to 12:
- Sum of 2: 1 way (1+1) – Probability: 1/36
- Sum of 3: 2 ways (1+2, 2+1) – Probability: 2/36 or 1/18
- Sum of 4: 3 ways (1+3, 2+2, 3+1) – Probability: 3/36 or 1/12
- Sum of 5: 4 ways (1+4, 2+3, 3+2, 4+1) – Probability: 4/36 or 1/9
- Sum of 6: 5 ways (1+5, 2+4, 3+3, 4+2, 5+1) – Probability: 5/36
- Sum of 7: 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) – Probability: 6/36 or 1/6
- Sum of 8: 5 ways (2+6, 3+5, 4+4, 5+3, 6+2) – Probability: 5/36
- Sum of 9: 4 ways (3+6, 4+5, 5+4, 6+3) – Probability: 4/36 or 1/9
- Sum of 10: 3 ways (4+6, 5+5, 6+4) – Probability: 3/36 or 1/12
- Sum of 11: 2 ways (5+6, 6+5) – Probability: 2/36 or 1/18
- Sum of 12: 1 way (6+6) – Probability: 1/36
To summarize, the probabilities of rolling sums with two dice vary, with the most common sum being 7 and the least common sums being 2 and 12. These probabilities showcase the interesting dynamics of chance and can be useful in various gaming strategies and mathematical exercises.