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The complement rule is a foundational concept in probability. To understand it, remember that the total probability of all possible outcomes must equal 1. This means if you know the probability of an event happening, you can easily find the probability of it not happening.
For an event A, the probability of the complement of A (A not happening) is given by: \(P(A^c) = 1 - P(A)\)
In the exercise, the complement rule was used for questions (b) and (d). For instance, question (b) required the probability that at least one has not driven while under the influence. This is the complement of the probability that all three have driven while under the influence. So, we used: \(P(\text{at least one has not driven}) = 1 - P(\text{all three have driven})\)
Similarly, in question (d), to find the probability that at least one has driven while under the influence, we used the complement of the probability that none of them have driven: \(P(\text{at least one has driven}) = 1 - P(\text{none have driven})\). This concept simplifies complex problems by utilizing the relationship between an event and its complement.

Events are independent if the occurrence of one event does not affect the probability of another. Understanding this concept is crucial when dealing with probabilities over multiple trials.
In the given exercise, the probabilities each 21- to 25-year-old has driven under the influence are assumed independent of one another. This means the chance one person has driven under the influence doesn’t change whether another person has or hasn’t.
For probability calculations involving multiple independent events, we multiply the probabilities of each individual event. This principle is applied in questions (a) and (c). In question (a), finding the probability that all three have driven while under the influence requires multiplying the probability for a single event, 0.29, three times: \(P(\text{all three}) = 0.29 \times 0.29 \times 0.29\). For question (c), the same idea is used with the probability of not having driven under the influence: \(P(\text{none have driven}) = 0.71 \times 0.71 \times 0.71\).
This simplicity and relevance make the concept of independent events a cornerstone of probability theory.

The multiplication rule helps in finding the probability that multiple independent events will occur together. As an extension of the concept of independent events, the multiplication rule states: \(P(A \text{ and } B) = P(A) \times P(B)\), provided A and B are independent.
In the exercise, this rule was vital in several calculations. For instance, to find the probability that all three individuals have driven while under the influence, we utilized: \(P(\text{all three}) = 0.29 \times 0.29 \times 0.29\), generating a final result of 0.024389. This application underlines how the multiplication rule magnifies the importance of independence between events.
Every student should understand that this rule only applies when events are independent. Misapplication can lead to errors, especially in real-world scenarios where dependencies might exist.

Event probability is the likelihood or chance that a specific outcome will occur. It ranges between 0 (impossible event) and 1 (certain event). The sum of probabilities of all possible outcomes of a random experiment always equals 1.
In the given problem, the probability of a single event (a person driving under the influence) is 0.29. The event probabilities are fundamental to calculating more complex probabilities involving multiple events.
When multiple individuals or trials are involved, as in questions (a) through (d), understanding how individual event probabilities aggregate (such as through multiplication for independent events) allows for solving larger-scale problems effectively. For instance, knowing \(P(\text{none have driven}) = 0.71 \times 0.71 \times 0.71\) or the previously discussed uses of the complement rule highlights how individual probabilities cohesively build up or influence cumulative event probabilities.