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In the context of probability theory, complementary events cover scenarios where one event is the opposite of another. For example, when discussing the probability of a U.S. resident not traveling to either Canada or Mexico, we are utilizing the concept of complementary events. The formula for finding the probability of a complement is:

• \( P(eg A) = 1 - P(A) \)

In the given exercise, we found that the probability of someone traveling to either Canada or Mexico, \( P(C \cup M) \), is 0.23. To find the probability that a person has not traveled to either country, we calculate its complement by subtracting \( P(C \cup M) \) from 1, resulting in 0.77. This simple alteration allows us to look at the flip side of the original event effectively.

The union of events in probability theory refers to the likelihood of one or more events occurring. When dealing with probabilities, the union of two events can be calculated using the formula:

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

This formula helps us determine the probability of either event A or event B, or both, happening. In our exercise, we calculated the probability that a U.S. resident has traveled to either Canada or Mexico, represented as \( P(C \cup M) \). By applying the union formula, we substituted the known values for the individual probabilities and the probability of traveling to both countries, which provided us a result of 0.23. This shows the extent of coverage across both events without over-counting the overlap.

Conditional probability is a key part of probability theory, dealing with the probability of an event occurring, given that another event has already occurred. While not directly included in every step of the given exercise, understanding conditional probability helps in deeper comprehension of probability problems. The formula generally used is:

• \( P(A | B) = \frac{P(A \cap B)}{P(B)} \)

This indicates how likely A is, provided B is known to happen. Although our main calculations didn't call for conditional probability explicitly, knowing it allows students to differentiate statistics based on additional criteria or different known probabilities. Understanding when and how to apply conditional probability can enhance problem-solving skills in more complex situations.

Probability rules provide the foundation for finding the probabilities of more complex events by evaluating simpler probabilities and relationships.

• **Sum Rule:** Helps in determining the probability of the union of two events.
• **Complement Rule:** Determines the probability of the complement of an event.
• **Multiplication Rule:** Often assists with finding intersections in probability.

Our exercise uses both the sum and complement rules extensively. For instance, we employ the sum rule to find the combined likelihood of traveling to either Canada or Mexico. Meanwhile, the complement rule provides a clear path to determine who has not traveled to either country. By mastering these foundational rules, one can effectively solve a wide array of probability problems with confidence.