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The sample space is a fundamental concept in probability, representing all possible outcomes of an experiment. For the digital communication channel transmitting four bits, each bit can either be received as distorted (D) or correct (C). This creates two possibilities for each bit.

When calculating the sample space for a series of bits, we consider every possible combination of these outcomes. Therefore, the sample space for four bits becomes a set with a total of \( 2^4 = 16 \) different combinations. Each combination represents one possible outcome of the experiment. Here are some examples:

• \( (C, C, C, C) \) - All bits are received correctly.
• \( (C, C, C, D) \) - Only the last bit is distorted.
• \( (D, D, D, D) \) - All bits are distorted.

These examples illustrate that the sample space is exhaustive and covers every potential result for the transmitted bits.

Mutually exclusive events are events that cannot happen simultaneously. In the context of our digital communication problem, let's see if the events \(A_i\) (where each \(A_i\) is the event that the \(i\)th bit is distorted) are mutually exclusive.

In this scenario, since each bit is independent, it means that one bit being distorted does not affect the others. Consequently, these events are not mutually exclusive because:

• More than one bit can be distorted at the same time.
• For instance, it's possible for both the first and second bits to be distorted simultaneously.

Thus, while evaluating digital communication bit errors, we find that these are independent events that can occur together rather than being mutually exclusive.

Complementary events are pairs of events where the occurrence of one event means the other cannot occur. In probability terms, they are two events that together encompass the entire sample space. If you take an event \( A_1 \), which represents the first bit being distorted, its complement \( A_1' \) is the scenario where this first bit is not distorted.

Using the example of our sample space:

• \( A_1' \) includes combinations like \( (C, C, C, C) \) and \( (C, C, D, D) \), where the first bit is not distorted.
• \( A_1' \) covers exactly the scenarios that \( A_1 \) does not cover.

Essentially, adding \( A_1 \) and \( A_1' \) together, you would obtain the whole sample space, ensuring that every possible event has been accounted for.

Intersection and union are operations in probability that allow us to combine different events in interesting ways.

The intersection \(A_1 \cap A_2 \cap A_3 \cap A_4\) refers to the event where all four bits are distorted. In this scenario, there is only one possible outcome: \( (D, D, D, D) \). This shows the conjunction of all individual events occurring together.

The union of events, like \((A_1 \cap A_2) \cup (A_3 \cap A_4)\), indicates cases where either the first two or the last two bits are distorted. This event covers multiple combinations:

• \( (D, D, C, C) \) - first two bits distorted.
• \( (C, C, D, D) \) - last two bits distorted.
• Other combinations where both conditions overlap, like \( (D, D, D, D) \).

By understanding intersections and unions, you can see how multiple conditions interplay to form complex probability scenarios.