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In probability theory, an event is any collection of outcomes from a probability trial or experiment. Simply put, it's something that can happen or not happen. For example, when you toss a coin, getting a 'head' is an event. Similarly, events can be more complex, like the arrival of a bus within the next 5 minutes.

Events are often denoted by capital letters such as \( A \), \( B \), or \( C \). Probabilities are numbers that measure the chance of these events occurring, ranging from 0 (the event will not occur) to 1 (the event will certainly occur).

When dealing with multiple events, we can explore their interactions. There are specific terms used in probability to describe such interactions, like the **union of events** and whether events are **independent or dependent**. These concepts help us compute probabilities in scenarios involving more than one event.

The union of two events \(A\) and \(B\) is the set of outcomes that are in either event, or both. In mathematical terms, it's expressed as \( A \cup B \).

To find the probability of the union \( P(A \cup B) \), we add the probabilities of the two events and subtract the probability that both events happen at the same time. This subtraction is necessary because adding the probabilities alone would count the overlapping outcomes twice. The formula is given by:

• Probability of the union: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

In our example, since \( P(A) = 0.20 \), \( P(B) = 0.30 \), and \( P(A \cap B) = 0.15 \), we substitute these values into the formula. Performing the arithmetic gives us \( P(A \cup B) = 0.20 + 0.30 - 0.15 = 0.35 \). This result represents the probability that either event \( A \) or event \( B \) occurs.

Understanding whether events are independent or dependent is crucial in probability. **Independent events** are those whose outcomes do not affect one another. For instance, flipping a coin and rolling a die are independent actions. The result of one does not change the probability of the other.

In contrast, **dependent events** have outcomes that affect each other. Imagine drawing a card from a deck, then drawing a second card without replacing the first. The probability of drawing a specific card the second time depends on what happened with the first draw.

In our exercise, knowing the probability of both \( A \) and \( B \) happening is critical. The event of both happening simultaneously, \( P(A \cap B) \), is considered in calculating their union. This aspect can indicate dependence if changes in one event influence the other. However, without additional context from the problem, we might not explicitly classify the events in the exercise as independent or dependent. Understanding these conditions aids in accurate probability computations.