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When rolling two dice, you are engaging in a simple yet fundamental experiment in probability theory. The collection of all possible results from this experiment is what we refer to as the **Sample Space**. Each die has six faces with numbers from 1 to 6. Thus, when we roll two dice together, there are multiple combinations that can happen.

To determine the sample space, you calculate the total number of outcomes by multiplying the number of faces on the first die by the number of faces on the second die, which is calculated as \(6 \times 6 = 36\). This tells us that there are 36 possible different outcomes when two dice are rolled.

• Each element in the sample space can be represented as a pair of numbers like (1,1), (1,2),...,(6,6), where the first number is from the first die and the second number from the second die.

Understanding the sample space is crucial for determining probabilities as it provides the foundation to assess where favorable outcomes exist within all possibilities.

To calculate probability, identifying **Favorable Outcomes** is essential; they are the outcomes that satisfy the event you are evaluating. In our dice rolling scenario, we are interested in finding the pairs whose sums are either 7 or 11.

• The sum 7 can be obtained from the following pairs: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). These six combinations illustrate how various numbers on the die can come together to total seven.
• On the other hand, to get a sum of 11, the pairs (5,6) and (6,5) fit the criteria. These are the only combinations that give us a sum of eleven.

Therefore, the total number of favorable outcomes in this scenario is 8, as calculated by adding the outcomes for sums of 7 and 11 together. These outcomes are what we need to determine the probability of the event occurring.

Now that we know the sample space and favorable outcomes, we can proceed to calculate the probability of our event. The **Probability Calculation** tells us how likely an event is to happen. It's determined by dividing the number of favorable outcomes by the total possible outcomes in the sample space.

Given our exercise, we have 8 favorable outcomes (where the sum of the numbers is either 7 or 11) out of a sample space of 36 total outcomes. Using this information, the probability is computed as follows:
\[ P(E) = \frac{8}{36} \]
By simplifying the fraction, we derive a final probability of
\[ P(E) = \frac{2}{9} \]

• This means that there's a \( \frac{2}{9} \) chance that the sum of numbers on two rolled dice will either be 7 or 11.

Such calculations are vital in probability theory as they allow us to quantitatively describe the likelihood of different outcomes, helping underpin more complex statistical and probabilistic analyses.