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In the realm of probability, a sample space is a fundamental concept. It's the set that contains every possible outcome from an experiment or an event.
For instance, if you're rolling a six-sided die, your sample space would simply be \(\{1, 2, 3, 4, 5, 6\}\) since these are all the possible numbers that could land face up. In the provided exercise, the sample space consists of \(A, B, C, \text{ and } D\).

• Every possible event, whether it happens or not, is part of the sample space.
• It is the comprehensive collection of all potential outcomes.

Understanding the sample space is important because it helps us define events and calculate probabilities effectively.

The complement of an event is an interesting and essential concept in probability. It consists of all outcomes that are not part of the specific event in question.
When we look at a sample space, the complement helps us see what is left over when one event has already occurred. Suppose your event is \(A\); thus, its complement, symbolized by \(A^c\), entails all other events in the sample space that are not \(A\). In our exercise, this means that \(A^c = \{B, C, D\}\).

• The principle of complementarity helps understand what might happen if a specific event does not.
• Probabilities of events and their complements generally add to 1, because one of them must occur.

Complementary events are a powerful tool as they simplify calculating probabilities, especially when it's easier to count the outcomes of the complement rather than the event itself.

Mutually exclusive events are events that cannot happen at the same time. They are an essential part of probability theory, assisting in understanding how different situations can unfold.
When events are mutually exclusive, the occurrence of one event means the others cannot occur simultaneously. In the exercise, if we consider the event \(A\) and its complement \(A^c\), they are mutually exclusive because if \(A\) happens, \(B, C, \text{ and } D\) do not, and vice versa.

• This mutual exclusivity is why we can perceive the sample space as having two exclusive events: \(A\) and \(A^c\).
• In simple terms, it means one event rules out the other completely.

Understanding mutually exclusive events is crucial, as it influences how we calculate probabilities. For mutually exclusive events, the probability that one or the other happens is simply the sum of their probabilities.