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The Complement Rule is a fundamental idea in probability theory. It assists in determining the probability that an event does not occur by providing a simple and efficient formula. In this rule, if you know the probability of an event happening, the probability of the event not happening (complement of the event) can easily be found.

Mathematically, the Complement Rule states:

• Let the probability of an event occurring be represented as \( P(A) \).
• The probability of the event not occurring is \( P(A^c) \), which is the complement of \( P(A) \).
• The equation is: \[ P(A^c) = 1 - P(A) \]

In our plumbing truck scenario, once we found the probability of at least one truck being available, we used the Complement Rule to determine the probability that neither truck was available. By subtracting \( 0.95 \) from \( 1 \), we found that there is a \( 0.05 \) (or 5%) chance that both trucks are out of service.

The Union of Events concept is often utilized to calculate the probability of at least one of multiple events occurring. It represents the culmination of "either-or" scenarios in probability theory. The formula for finding the probability of the union of two events \( A \) and \( B \) helps in understanding how multiple outcomes can interact with one another.

Here is the formula for the union of two events:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

This formula accounts for the overlap between \( A \) and \( B \), denoted by \( P(A \cap B) \), which represents the probability of both events occurring simultaneously. By subtracting \( P(A \cap B) \), we ensure that this overlapping probability isn't counted twice.

In the exercise with the two trucks:

• \( P(A) \) is the probability the first truck is available, which is \( 0.75 \).
• \( P(B) \) is the probability the second truck is available, which is \( 0.50 \).
• We already know \( P(A \cap B) = 0.30 \).

By substituting these values into the formula, we calculated \( P(A \cup B) = 0.95 \), representing the chance that at least one of the trucks is operational.

In probability theory, events are termed 'independent' when the occurrence of one does not affect the occurrence of the other. Understanding whether events are independent or dependent is crucial to solving many probability-related problems.

For independent events:

• The probability of both events \( A \) and \( B \) occurring simultaneously is the product of their probabilities: \[ P(A \cap B) = P(A) \times P(B) \]

If the given probabilities do not satisfy this relationship, the events are dependent.

In the truck problem:

• We have \( P(A \cap B) = 0.30 \).
• For true independence, we would expect \( P(A) \times P(B) = 0.75 \times 0.50 = 0.375 \).

Since \( P(A \cap B) = 0.30 \), not \( 0.375 \), these events are dependent. Thus, the availability of one truck has some effect on the availability of the other, indicating dependency rather than independence.