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. **Understand the Total Possible Outcomes** When you roll 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. **Determine the Favorable Outcomes** You want the sum of the two dice to be greater than 7. The possible sums greater than 7 are **8, 9, 10, 11, and 12**. Let's list all the combinations that result in these sums: - **Sum of 8:** - (2, 6) - (3, 5) - (4, 4) - (5, 3) - (6, 2) *Total combinations: 5* - **Sum of 9:** - (3, 6) - (4, 5) - (5, 4) - (6, 3) *Total combinations: 4* - **Sum of 10:** - (4, 6) - (5, 5) - (6, 4) *Total combinations: 3* - **Sum of 11:** - (5, 6) - (6, 5) *Total combinations: 2* - **Sum of 12:** - (6, 6) *Total combinations: 1* **Adding these up:** \[ 5 + 4 + 3 + 2 + 1 = 15 \text{ favorable outcomes} \] ### 3. **Calculate the Probability** The probability \( P \) of rolling a sum greater than 7 is the number of favorable outcomes divided by the total number of possible outcomes: \[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{15}{36} \] This fraction can be simplified by dividing both the numerator and the denominator by 3: \[ \frac{15 \div 3}{36 \div 3} = \frac{5}{12} \] ### **Final Answer:** The probability of rolling a total greater than 7 with two six-sided dice is **\(\frac{5}{12}\)**, which is approximately **41.67%**.