The sample space is:
| 1 + 1 = 2 | 1 + 2 = 3 | 1 + 3 = 4 | 1 + 4 = 5 | 1 + 5 = 6 | 1 + 6 = 7 |
| 2 + 1 = 3 | 2 + 2 = 4 | 2 + 3 = 5 | 2 + 4 = 6 | 2 + 5 = 7 | 2 + 6 = 8 |
| 3 + 1 = 4 | 3 + 2 = 5 | 3 + 3 = 6 | 3 + 4 = 7 | 3 + 5 = 8 | 3 + 6 = 9 |
| 4 + 1 = 5 | 4 + 2 = 6 | 4 + 3 = 7 | 4 + 4 = 8 | 4 + 5 = 9 | 4 + 6 = 10 |
| 5 + 1 = 6 | 5 + 2 = 7 | 5 + 3 = 8 | 5 + 4 = 9 | 5 + 5 = 10 | 5 + 6 = 11 |
| 6 + 1 = 7 | 6 + 2 = 8 | 6 + 3 = 9 | 6 + 4 = 10 | 6 + 5 = 11 | 6 + 6 = 12 |
This is a conditional probability. The cases where the first roll is a 6 are highlighted below:
| 1 + 1 = 2 | 1 + 2 = 3 | 1 + 3 = 4 | 1 + 4 = 5 | 1 + 5 = 6 | 1 + 6 = 7 |
| 2 + 1 = 3 | 2 + 2 = 4 | 2 + 3 = 5 | 2 + 4 = 6 | 2 + 5 = 7 | 2 + 6 = 8 |
| 3 + 1 = 4 | 3 + 2 = 5 | 3 + 3 = 6 | 3 + 4 = 7 | 3 + 5 = 8 | 3 + 6 = 9 |
| 4 + 1 = 5 | 4 + 2 = 6 | 4 + 3 = 7 | 4 + 4 = 8 | 4 + 5 = 9 | 4 + 6 = 10 |
| 5 + 1 = 6 | 5 + 2 = 7 | 5 + 3 = 8 | 5 + 4 = 9 | 5 + 5 = 10 | 5 + 6 = 11 |
| 6 + 1 = 7 | 6 + 2 = 8 | 6 + 3 = 9 | 6 + 4 = 10 | 6 + 5 = 11 | 6 + 6 = 12 |
Given these 6 cases, there are 4 in which the sum is greater than 8. So, P(sum ≥ 8 | first die is 6) = 4/6 = 2/3.