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In probability theory, events are outcomes or occurrences that result from a random experiment. These events are key to calculating probabilities. For example, if you drop a coin, the events are 'heads' or 'tails'. In our scenario, the events are 'Ann is hired' and 'Bev is hired'. These events are often represented by capital letters like \( A \) for Ann and \( B \) for Bev.

Each event has a probability, which is a number between 0 and 1 indicating the likelihood of the event occurring. The higher the probability, the more likely the event will occur. Sometimes, multiple events can be combined or manipulated using set operations to produce new events, which is what we explore in our examples.

The complement of a set refers to all the outcomes that are not part of the particular event. In the context of our problem, the complement of the event \( B \), denoted as \( B^c \), includes all outcomes where Bev is not hired.

The complement is important because it helps us understand what happens when an event does not occur. For instance, when we say Ann is hired but Bev is not, we are effectively taking the intersection of Ann's hiring event \( A \) with the non-hiring of Bev \( B^c \). This is mathematically expressed as \( A \cap B^c \).

By defining complements, we get a complete picture of all possible outcomes, aiding in better decision-making and probability calculation.

The intersection of two sets includes all outcomes shared by both sets. When dealing with probability events, the intersection \( A \cap B \) represents the situation where both Ann and Bev get hired.

However, if you want the scenario where Ann is hired but Bev is not, you focus on the intersection between \( A \) and \( B^c \), which gives us the outcomes common to these conditions. It is useful for specificity, allowing us to pinpoint exactly when two conditions are true at the same time.

Understanding the intersection is crucial, especially when analyzing events that require the fulfillment of multiple conditions simultaneously.

The union of sets in probability represents all the outcomes that belong to at least one of the sets. In our example, the union \( A \cup B \) indicates the scenario where at least one of Ann or Bev is hired.

This kind of set operation is broad, capturing multiple potential outcomes in one swoop. It is vital when you're interested in any of multiple events occurring. This means if even one condition holds true, the union captures it.

Additionally, for the event where exactly one of them is hired, the union \( (A \cap B^c) \cup (B \cap A^c) \) is used. This ensures that either only Ann is hired or only Bev is hired, but not both. Understanding the union helps in considering all possibilities and ensuring inclusive probabilistic scenarios.