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Probability theory is the mathematical framework used to quantify randomness and uncertainty. It provides tools for analyzing random events and understanding their likelihoods. This foundational concept is essential in various fields such as finance, insurance, and science.

In probability theory, events are outcomes or sets of outcomes from a random process. The probability of an event represents the likelihood that it will occur. The values range from 0 to 1, where 0 signifies impossibility and 1 signifies certainty.

Understanding probability theory helps us make informed decisions. It enables us to predict future occurrences and to assign meaningful levels of confidence to those predictions. By mastering this theory, we can evaluate risks and opportunities more accurately.

Set operations are fundamental in probability theory as they allow us to manipulate and understand the relationships between different events. These operations include union, intersection, complement, and difference, among others. In the context of probability, we often describe events using these set operations.

• **Union**: The union of two sets, denoted as \(A \cup B\), contains all elements that are in either set \(A\) or set \(B\) or both.
• **Intersection**: The intersection, \(A \cap B\), includes only the elements found in both \(A\) and \(B\).
• **Complement**: The complement of a set, such as \(\bar{A}\), includes all elements that are not in \(A\).
• **Difference**: The difference is represented by \(A - B\) or \(A \setminus B\), which includes elements that belong to \(A\) but not \(B\).

Set operations provide a language for describing complex relationships in a structured way. Mastery of these concepts is crucial for solving advanced probability problems, like finding the probability of a symmetric difference as explored in this exercise.

Event probability refers to the chance of a specific event occurring. It's a core concept in probability theory that stems from understanding individual events and how they relate to each other.

To calculate event probability, we first define the total number of possible outcomes in the sample space, which is the set of all possible occurrences. Then, by examining the number of ways a particular event can happen, we calculate its probability as the ratio of favorable outcomes to the total outcomes.

For example, if you roll a fair six-sided die, the probability of rolling a four is \(\frac{1}{6}\) since there is 1 favorable outcome (rolling a four) out of 6 possible outcomes. This idea is expanded further when dealing with more complex events, where the application of set operations, like union and intersection, comes into play. This helps in calculating the combined or joint probabilities of multiple events.

The concepts of union and intersection are key to calculating probabilities involving multiple events. They allow us to understand combined probabilities when considering two or more events together.

The **union** of two events \(A\) and \(B\) represents all outcomes that are in either one of the events or both. This is calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]This formula helps avoid double counting outcomes that appear in both \(A\) and \(B\).

Meanwhile, the **intersection** of events, \(A \cap B\), accounts for only those outcomes shared by both events. Understanding the values of these probabilities is often challenging but essential in solving problems like the one presented in the exercise.

These concepts are foundational for grasping more advance topics such as symmetric difference. The ability to effectively use intersection and union calculations clarifies the relationships between events and supports accurate probability assessments.