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The term 'sample space' is fundamental to understanding probability. It encapsulates all the possible outcomes that can result from a particular experiment or activity. When a die is rolled, as in our example, the sample space consists of all the sides that might land face up. Since a standard die has six faces, numbered 1 through 6, the sample space for a single roll is \( S = \{1, 2, 3, 4, 5, 6\} \). Each of these outcomes is equally likely if the die is fair.

Understanding and listing the sample space is crucial because it sets the stage for determining probabilities. It's a way to visualize all the potential results before we hone in on specific outcomes of interest. Without this comprehensive list, it would be difficult to assess the likelihood of any particular event.

Probabilities are directly tied to the concept of 'favourable outcomes,' which refers to those outcomes within the sample space that satisfy a given condition. In the scenario of rolling a die and wanting a number greater than 3, we focus on results that fit this criterion. Hence, our favourable outcomes are 4, 5, and 6.

The identification of favourable outcomes is an exercise in alignment with the specific event we are interested in. Not all outcomes in the sample space may be relevant to our calculation. In this case, since only the numbers 4, 5, and 6 meet the condition, they are the ones we count. The precision with which we identify these outcomes helps us to accurately assess the probability of the event occurring. Identifying and understanding which outcomes of an experiment are considered favourable is the stepping-stone to calculating probability, and ensuring these are correctly listed is key to an accurate calculation.

Once we have our sample space and favourable outcomes, we use the probability formula to pinpoint the chance of a specific event. The formula is concise but powerful: \(\text{Probability} = \frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Outcomes}}\). To apply this formula, simply plug in numbers from our example: with three favourable outcomes (4, 5, 6) and six possible outcomes in total, the probability is \(\frac{3}{6}\), which simplifies to 0.5.

The beauty of this formula lies in its wide applicability, as it's used for any event where outcomes are equally likely. Understanding how to apply it is crucial in the world of probability because it translates abstract concepts like 'chance' and 'likelihood' into concrete numbers. It encapsulates a core principle of probability — every event's probability is a ratio that compares the count of favourable cases to the total number of equally possible cases.