If a pair of dice, one green and one red, is rolled, what is the probability that the sum of the two dice is a number between 5 and 10, inclusive?
There are $6\times 6=36$ possible outcomes when the two dice are rolled. We want to count the number of outcomes that sum to a number between 5 and 10, inclusive. We see that 5 and 10 cannot be achieved, as the maximum sum is $6+6=12$. So, we need to count the number of ways to achieve a sum of 6, 7, 8, or 9.
For a sum of 6, there are five possible outcomes: (1,5), (2,4), (3,3), (4,2), and (5,1).
For a sum of 7, there are six possible outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
For a sum of 8, there are five possible outcomes: (2,6), (3,5), (4,4), (5,3), and (6,2).
For a sum of 9, there are four possible outcomes: (3,6), (4,5), (5,4), and (6,3).
Therefore, there are $5+6+5+4=20$ total outcomes that sum to a number between 5 and 10, inclusive.
The probability is therefore $\dfrac{20}{36}=\boxed{\dfrac{5}{9}}$.
To find the probability of rolling a sum between 5 and 10 (inclusive) on the two dice, we need to determine the favorable outcomes and the total possible outcomes.
Let's first determine the favorable outcomes:
The pairs (ordered) that have a sum between 5 and 10 are:
(1, 4), (2, 3), (3, 2), (4, 1), (2, 4), (3, 3), (4, 2), (3, 4), (4, 3), and (4, 4).
So, there are 10 favorable outcomes.
Now, let's determine the total possible outcomes:
Since we have a green die and a red die, each die has 6 possible outcomes (numbers 1 to 6). Therefore, the total number of possible outcomes is 6 × 6 = 36.
Now, we can calculate the probability:
Probability = favorable outcomes / total possible outcomes
Probability = 10 / 36
Simplifying, we find:
Probability = 5 / 18
So, the probability of rolling a sum between 5 and 10 (inclusive) on the two dice is 5/18.