91Ó°ÊÓ

In probability theory, the term \(P(A \cup B \cup C)\) refers to the probability that at least one of the events \(A\), \(B\), or \(C\) occurs. This is called the union probability of the events. When calculating this probability, one needs to account for the possibility of overlap between the events.
To visualize this, imagine each event as a circle in a Venn diagram. The probability of the union covers all areas within these circles. Using the Inclusion-Exclusion Principle is crucial here to ensure correct calculation, especially because events may overlap or occur simultaneously. Identifying intersections will be key to determining the union probability accurately.
Each step in this calculation ensures that we appropriately include all events and correct for overlaps. This aim is to determine how likely it is for at least one of these events to happen.

The Probability Addition Rule helps us start our calculation process for union probability by considering each event's individual probabilities. Initially, we add these probabilities together: \(P(A) + P(B) + P(C)\). This first sum implies that each event is independent of overlaps, hence it sums up each circle's area in the Venn diagram.
This basic step assumes no interaction between events, meaning that each event is calculated as if it's unrelated to the others. At this point, event overlaps are not yet considered; thus, they are potentially counted multiple times if they exist.
Understanding this addition is our starting point, but it doesn't give an entirely accurate result until we handle intersections in the subsequent steps.

To correct the over-counting from the Probability Addition Rule, we must subtract the probability of pairwise intersections: \(P(A \cap B)\), \(P(B \cap C)\), and \(P(C \cap A)\). Each of these terms corresponds to where two events intersect, meaning they happen at the same time. In a Venn diagram, these are the areas where two circles overlap.
The subtraction acknowledges that the overlap regions have been counted twice so far—once with each event's probability. By removing these double-counted portions, we refine our union probability calculation.
For instance, \(P(A \cap B)\) excludes the count of both \(A\) and \(B\) when they simultaneously occur. It reflects the reality of events intersecting in practical scenarios. This precise adjustment ensures overlaps aren't wrongly inflated in our probability estimate.

Finally, we must address the situation where all three events, \(A\), \(B\), and \(C\), occur simultaneously, represented by \(P(A \cap B \cap C)\). When subtracting pairwise intersections previously, we inadvertently subtracted the triple intersection more than once.
Adding back \(P(A \cap B \cap C)\) corrects this oversight. It ensures that the area where all three circles overlap on a Venn diagram is counted just once in the final probability.
This step reflects real-world instances where all events occur together. It verifies that this unique scenario of full overlap is accounted for appropriately. Thus, the triple intersection ensures our calculations precisely depict the actual likelihood across all considered events.