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When you roll two dice, the resulting face values can be combined in several ways to create different sums. These sums are not equally likely to occur because some combinations can be achieved in more ways than others. For instance, there is only one way to get a sum of 2 (by rolling a one on both dice), whereas there are six ways to get a sum of 7 (1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1).

The analysis of these outcomes is the bedrock of understanding probability with dice rolls. With the knowledge of each possible outcome, in our case a sum ranging from 2 to 12, we can begin to calculate the probabilities of each sum occurring. It’s important to systematically list out each pair of rolls to ensure none are missed, and to grasp why some sums appear more frequently in outcomes than others.

The process of calculating probabilities involves determining the likelihood of specific outcomes. In the context of dice rolling, the steps taken typically include identifying all possible outcomes, determining the frequency of the outcome of interest, and then using the classic probability formula.The formula for calculating the probability of an event is:\[ P(Event) = \frac{{number\ of\ favorable\ outcomes}}{{total\ number\ of\ possible\ outcomes}} \]In the dice example, the favorable outcomes for a given sum (like a sum of 7) are counted, and this count is then divided by the total number of possible outcomes (36, since each die has 6 faces and 6 x 6 = 36). Remember to reduce fractions to their simplest form to express the probability in the most understandable way.

The concept of discrete probability distribution comes into play when we deal with a finite set of possible outcomes, such as the sums obtained from rolling two dice. Each outcome has a probability associated with it, and together, these probabilities form a distribution.A discrete probability distribution is characterized by a list of possible values of a discrete random variable (like our sum of dice, X) and the probability associated with each value. It's discrete because the outcomes are clearly separated (you cannot roll a 2.5 with a dice).The probabilities must satisfy two conditions: each probability is between 0 and 1, and the sum of all probabilities equals 1. By calculating the probabilities for each potential sum, we effectively create this distribution for the sums of our rolled dice. It's a valuable tool for understanding the likelihood of different events occurring in our random experiment of rolling dice.