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When we talk about sample space in probability, we mean the set of all possible outcomes of an experiment. In the exercise, the experiment involves rolling two six-sided dice. Each die can land on a number from 1 to 6. This means that for two dice, combinations of these results form what's known as the **sample space**.

For this particular experiment with two dice, the sample space consists of all possible pairs \(n, m\) where \(n\) and \(m\) range from 1 to 6. To visualize it, think of a grid with 36 points, each representing an outcome like (1,1), (1,2), (1,3), ..., (6,6).

From knowing the sample space, we can make systematic and informed decisions about how various probabilities relate to different **events**. The size of the sample space is often the first step in solving probability problems, providing a complete picture of what could happen. Knowing this sets the stage for understanding the events, like the appearance of a "4" on at least one die.

An event is a specific outcome or set of outcomes that we're interested in when we carry out an experiment. In the context of the dice-roll problem, the event of interest is that a "4" appears on at least one die.

To identify such an event, we look for all outcomes in the sample space where either one or both numbers in the pair are 4. Essentially, it means selecting all pairs \( (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (1,4), (2,4), (3,4), (5,4), (6,4) \).

This event can be visualized as a subset of the full sample space. While the entire sample space contains 36 possibilities, the event we're looking for specifically focuses only on those pairs where the condition of having a "4" is met. Understanding events helps in calculating probabilities by filtering out relevant outcomes from the broader sample space.

Counting outcomes is a fundamental skill in probability that helps determine the number of favorable events. By doing this, we can calculate probabilities more efficiently.

To count outcomes for the event "4 appears on at least one die," we begin by selecting outcomes from the sample space that match our condition. As previously mentioned, there are 11 such pairs: (4,1), (4,2), ... , (6,4).

By identifying these specific outcomes, we can see how many times our event occurs out of the total possible outcomes, which is a key step in finding the probability of the event. To find the probability, divide the number of favorable outcomes (11 outcomes) by the total number of outcomes in the sample space (36 outcomes). This results in a probability of \(\frac{11}{36}\).

Mastering the ability to count and recognize outcomes within sample spaces is crucial for solving more complex probability problems. It builds the foundation for more advanced concepts like independent and dependent events, as well as conditional probability.