How do you calculate the probability of rolling a total greater than 7 with two six-sided dice?

To calculate the probability of rolling a total greater than 7 with two six-sided dice, you can follow these steps: ### 1. **Determine the Total Number of Possible Outcomes** When rolling two six-sided dice, each die has 6 faces. Therefore, the total number of possible outcomes is: \[ 6 \text{ (faces on Die 1)} \times 6 \text{ (faces on Die 2)} = 36 \text{ possible outcomes} \] ### 2. **Identify the Favorable Outcomes (Sum Greater Than 7)** Next, list all the possible combinations that result in a sum greater than 7. Here's a breakdown by each possible sum greater than 7: - **Sum of 8:** - (2, 6) - (3, 5) - (4, 4) - (5, 3) - (6, 2) *Total: 5 combinations* - **Sum of 9:** - (3, 6) - (4, 5) - (5, 4) - (6, 3) *Total: 4 combinations* - **Sum of 10:** - (4, 6) - (5, 5) - (6, 4) *Total: 3 combinations* - **Sum of 11:** - (5, 6) - (6, 5) *Total: 2 combinations* - **Sum of 12:** - (6, 6) *Total: 1 combination* **Adding them up:** \[ 5 (\text{for 8}) + 4 (\text{for 9}) + 3 (\text{for 10}) + 2 (\text{for 11}) + 1 (\text{for 12}) = 15 \text{ favorable outcomes} \] ### 3. **Calculate the Probability** Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes. \[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}} = \frac{15}{36} \] This fraction can be simplified by dividing both the numerator and the denominator by 3: \[ \frac{15}{36} = \frac{5}{12} \approx 0.4167 \] ### **Conclusion** The probability of rolling a total greater than 7 with two six-sided dice is: \[ \boxed{\dfrac{5}{12} \text{ or approximately } 41.67\%} \]