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In probability theory, the concept of a sample space is fundamental. The sample space represents all possible outcomes of an experiment. Each outcome is an element of this space.

For example, when four bits are transmitted, and each bit can either be distorted (D) or not distorted (N), the sample space consists of all possible sequences of these four bits. We denote this as:

• \( \{DDDD, DDDN, DDND, DDNN, DNDD, DNDN, DNND, DNNN, NDDD, NDDN, NDND, NDNN, NNDD, NNDN, NNND, NNNN\} \)

This means the total number of different outcomes is \(2^4 = 16\), because each of the four bits has two possible states.

Understanding the sample space is essential because it lays the foundation for determining the likelihood of various events.

Events are described as mutually exclusive if they cannot occur simultaneously. In other words, the occurrence of one event excludes the possibility of another.

In our four-bit transmission example, the events \(A_i\), which denote the distortion of individual bits, are **not** mutually exclusive. Multiple bits can be distorted at the same time.

This indicates that the events can overlap. For example, if event \(A_1\) means the first bit is distorted, and \(A_2\) means the second bit is distorted, they can both happen concurrently in an outcome like \(DDNN\). Therefore, these events overlap and are not mutually exclusive.

In probability, the complement of an event represents all outcomes not included in the event. It covers everything in the sample space that the event does not.

Taking the example of event \(A_1\), where the first bit is distorted, its complement \(A_1'\) consists of all outcomes where the first bit is not distorted. The complement event \(A_1'\) for our four-bit situation includes:

• \( \{NDDD, NDDN, NDND, NDNN, NNDD, NNDN, NNND, NNNN\} \)

Understanding complements is crucial, especially when calculating probabilities, as their probabilities add up to 1. This is particularly useful for determining the probability that at least one event occurs by using its complement.

The intersection and union are operations that allow us to evaluate and understand events involving two or more sets.

The intersection, denoted as \(A \cap B\), consists of outcomes that are common to both events \(A\) and \(B\). In our scenario, the intersection \(A_1 \cap A_2 \cap A_3 \cap A_4\) represents all bits being distorted, giving the single outcome \( \{DDDD\} \).

On the other hand, the union, \(A \cup B\), includes outcomes that belong to either of the events or both. For example, \((A_1 \cap A_2) \cup (A_3 \cap A_4)\) includes cases where either the first two bits, the last two bits, or both are distorted. This results in outcomes \( \{DDDD, NDDD, DDDN, DDND, DDNN, NDND\} \).

These operations are foundational in setting up complex event probabilities and comprehending how events correlate.