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Event operations are fundamental when dealing with probability theory and involve manipulating sets representing various outcomes. In our context, the events refer to the completion state of power plants. We can use three primary operations:

• Union (\(\cup\)): Combines multiple events and includes any situation where at least one of the events occurs. For example, the event \(A_1 \cup A_2\) signifies that either plant 1 or plant 2, or both, are completed.


• Intersection (\(\cap\)): Identifies scenarios where all considered events occur simultaneously. For instance, the event \(A_1 \cap A_2\) means both plant 1 and plant 2 are completed.


• Complementation: Represents outcomes where a specific event does not happen. Denoted as \(A^c\), it includes all situations except those in \(A\). If \(A_1^c\) is considered, it implies that plant 1 is not completed.

These tools allow us to describe complex real-world events using a concise mathematical language.

Venn diagrams offer a visual way to represent relationships between different events. In probability and set theory, they serve as a useful tool to visualize operations like unions, intersections, and complements.
A typical Venn diagram involves circles within a rectangle, each circle representing a unique event. Overlapping areas of these circles show intersections where events coincide, while distinct circles indicate exclusive occurrences.
For instance, if we represent events \(A_1, A_2,\) and \(A_3\) in a Venn diagram, the area overlapping between all three circles represents \(A_1 \cap A_2 \cap A_3\), where all plants are completed. Conversely, any region where at least one circle exists without overlap implies the union \(A_1 \cup A_2 \cup A_3\) – a situation where at least one plant is completed.
This graphical representation aids in understanding event operations intuitively, especially when determining complex combinations of outcomes.

Set theory is foundational in understanding events and operations within probability. Events in probability are essentially sets of outcomes, and set theory provides the tools to handle these efficiently.
Sets allow us to encapsulate information like the states of plant completion. For example, set \(A_1\) includes all possible outcomes where the plant at site 1 is completed. Meanwhile, operations such as union and intersection are direct applications of set theory principles.
Using notation like \(A_1 \cup A_2\) or \(A_1 \cap A_2\), we leverage set theory to describe complex scenarios neatly. Set complements, like \(A_1^c\), further enrich our ability to manage and express what's possible or not within these event combinations.
Ultimately, set theory underpins our approach to event operations, making it possible to translate real-world scenarios into mathematical expressions that are manageable and understandable.

Union and intersection are pivotal concepts in probability as they define how multiple events relate to one another. The union of events, denoted as \(A \cup B\), represents situations where either event \(A\), event \(B\), or both occur. This is crucial for scenarios like determining if at least one power plant completes its construction in time.
Conversely, the intersection, indicated as \(A \cap B\), focuses on cases where both events occur simultaneously. This is seen in probabilities where we need to know if all power plants meet their respective deadlines.
Understanding these operations assists in breaking down complex problems. For example, combining events with multiple unions and intersections allows us to specify conditions like completing exactly one plant or none at all.
These foundational concepts help translate conditions into mathematical language, which can then be analyzed with precision and clarity, serving as a cornerstone in probability theory.