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Understanding how to calculate outcomes of a dice roll is fundamental in the study of probability. Each die in a pair is an independent object, and the total outcomes can be found through simple multiplication of each die's possibilities. Rolling a single six-sided die, you have precisely 6 potential outcomes, numbered 1 to 6.

When rolling two dice, the number of outcomes for the first die (6 possibilities) multiplies with the number of outcomes for the second (also 6 possibilities), resulting in a total of 36 distinct possible outcomes. These outcomes range from the pair (1,1), which is the lowest, to (6,6), the highest. Each outcome is represented as a pair of numbers, where the first number corresponds to the outcome on the first die, and the second number represents the outcome on the second die.

It's important to consider this concept of independence and multiplication when approaching any problem involving multiple rolling dice, as it forms the basis of most probability calculations in such scenarios.

Probability calculations help us determine the likelihood of a particular event occurring out of all possible events. To find the probability of a certain outcome when rolling two dice, you divide the number of ways that outcome can occur by the total number of possible outcomes. The probability formula used within the context of dice rolling is:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
For instance, there is only one way to roll a sum of 12 鈥� rolling two sixes 鈥� making the probability of this event:
\[ P(\text{sum of 12}) = \frac{1}{36} \].

Conversely, there are six different ways to roll a sum of 7, giving it a higher probability:
\[ P(\text{sum of 7}) = \frac{6}{36} = \frac{1}{6} \].

Thus, the probability of rolling a sum of 7 is much greater than that of rolling a sum of 12. This forms the core of probability calculations: counting the favorable outcomes and placing them in relation to the total number of outcomes.

Combinatorics is a mathematical concept that deals with counting, arranging, and combination of elements in sets, particularly with finite sets. It is at the heart of calculating probabilities with dice. When we evaluate the number of ways to achieve a certain sum with two dice, we're using principles of combinatorics.

In our exercise, we've employed basic combinatorial reasoning to determine that there's only one combination to get a sum of 12, and six combinations to get a sum of 7. It's crucial to remember that the order of elements in the pairs for dice outcomes does matter, because a roll of (1,6) is different from a roll of (6,1), even though they result in the same sum. Therefore, when counting the combinations for sums, each unique pairing counts independently.

The more complex the dice problem, the more sophisticated combinatorial techniques we may need to use, such as permutations and variations. Coupled with a solid understanding of the fundamental counting principle, combinatorics enables us to tackle a wide variety of probability problems in a structured and methodical way.