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

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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:**
   - Each die has 6 sides, so when rolling two dice, the total number of possible outcomes is $$6 \times 6 = 36$$.

2. **Identify the Favorable Outcomes (Sums Greater Than 7):**
   - You need to find all the combinations of the two dice that add up to 8, 9, 10, 11, or 12.
   
   Here's a breakdown:

| **Sum** | **Possible Combinations**              | **Number of Combinations** |
   |---------|---------------------------------------|----------------------------|
   | 8       | (2,6), (3,5), (4,4), (5,3), (6,2)     | 5                          |
   | 9       | (3,6), (4,5), (5,4), (6,3)            | 4                          |
   | 10      | (4,6), (5,5), (6,4)                    | 3                          |
   | 11      | (5,6), (6,5)                           | 2                          |
   | 12      | (6,6)                                  | 1                          |
   
   - **Total Favorable Outcomes:** $$5 + 4 + 3 + 2 + 1 = 15$$

3. **Calculate the Probability:**
   - **Probability Formula:** 
     $$
     \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}
     $$
   - Plugging in the numbers:
     $$
     \text{Probability} = \frac{15}{36} = \frac{5}{12} \approx 0.4167
     $$
   
   - **Interpretation:** There's approximately a 41.67% chance of rolling a total greater than 7 with two six-sided dice.

4. **Simplify the Fraction (Optional):**
   - The fraction $$\frac{15}{36}$$ can be simplified by dividing both the numerator and the denominator by 3:
     $$
     \frac{15 \div 3}{36 \div 3} = \frac{5}{12}
     $$
   
5. **Conclusion:**
   - The probability of rolling a sum greater than 7 with two six-sided dice is $$\frac{5}{12}$$ or approximately 41.67%.

**Visual Representation:**

You can also visualize all possible outcomes using a table:

| Die 1 \ Die 2 | 1 | 2 | 3 | 4 | 5 | 6 |
|---------------|---|---|---|---|---|---|
| **1**         | 2 | 3 | 4 | 5 | 6 | 7 |
| **2**         | 3 | 4 | 5 | 6 | 7 | 8 |
| **3**         | 4 | 5 | 6 | 7 | 8 | 9 |
| **4**         | 5 | 6 | 7 | 8 | 9 | 10|
| **5**         | 6 | 7 | 8 | 9 | 10| 11|
| **6**         | 7 | 8 | 9 | 10| 11| 12|

- **Favorable sums (8, 9, 10, 11, 12)** are highlighted with counts above.

This table helps visualize why there are 15 favorable outcomes out of 36 possible.
The probability of rolling a total greater than 7 with two six-sided dice is $$\frac{5}{12}$$ or approximately 41.67%.

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

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