You roll two dice. What is the probability that the sum of the dice is greater than 8 and that one of the dice shows a 6?

To find the probability that the sum of two dice is greater than 8 and that one of the dice shows a 6, we first determine the total number of possible outcomes when rolling two dice, which is 36 (since each die has 6 sides, and there are 6 * 6 = 36 possible combinations).

Next, we find the number of outcomes where one of the dice shows a 6. There are 6 possible outcomes where one die shows a 6, and the other die can show any number from 1 to 6. So, there are 6 * 6 = 36 outcomes where one die shows a 6.

Then, we find the number of outcomes where the sum of the dice is greater than 8. These outcomes include (2,7), (3,6), (3,7), (4,5), (4,6), (4,7), (5,4), (5,5), (5,6), (5,7), (6,3), (6,4), (6,5), (6,6), (6,7), (7,2), (7,3), (7,4), (7,5), (7,6), and (7,7). There are 21 such outcomes.

Now, we need to find the number of outcomes where both conditions are met: the sum is greater than 8 and one die shows a 6. There are 4 such outcomes: (2,7), (3,6), (4,5), and (5,6).

Therefore, the probability of rolling two dice and getting a sum greater than 8 while one of the dice shows a 6 is 4/36, which simplifies to 1/9.