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The concept of a sample space is foundational in understanding probability theory. It is defined as the set of all possible outcomes of a random experiment. For instance, if we're looking at the roll of a die, the sample space would be \( S = \{1, 2, 3, 4, 5, 6\} \), comprising each number that could possibly come up when the die is rolled.

In the given exercise, the sample space for the experiment is represented by \( S = \{a, b, c, d, e\} \). Understanding this concept is crucial because it sets the stage for determining probabilities of various events – an event being a subset of the sample space. In probability, we are often interested in the likelihood of one specific event occurring out of the full set of possible events within our sample space.

The complement of an event in probability is a key concept, which refers to all the outcomes in the sample space that are not included in the event. Symbolically, if we have an event \( A \), its complement is denoted by \( A^c \), and it encompasses everything in the sample space that is not \( A \).

With reference to the exercise, for event \( E = \{a, b\} \), the complement \( E^c \), includes every element from the sample space \( S \) that isn't in \( E \), resulting in \( E^c = \{c, d, e\} \). This notion is fundamental as it helps in calculating probabilities related to 'not happening' of an event and is deeply interconnected with the concept of set theory in probability, as it essentially deals with elements belonging to different sets.

Set theory forms the backbone of probability because it provides a structured way of dealing with events, which are themselves sets. Main operations in set theory, such as unions, intersections, and complements, are paralleled in probability operations on events. A key principle of set theory in probability is the concept of 'mutually exclusive' events, where two events cannot occur simultaneously – their intersection is an empty set.

In the exercise, while the original events \( E \) and \( F \) are mutually exclusive, their complements are not, as evident from their intersection \( E^c \cap F^c = \{e\} \), which contains the element \( e \). This shows that within the set theory framework, the complement does not necessarily preserve the property of mutual exclusivity between the original sets.