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The Principle of Inclusion-Exclusion is a pivotal concept in probability theory, especially when dealing with the probability of the union of events. It is a counting technique used to find the probability of the union of multiple events by first adding the probabilities of each event separately. However, if we stop there, we have counted some of the outcomes more than once - those that occur in the intersection of events. To correct this, we subtract the probabilities of each pairwise intersection among the events.

One might ask, 'What if an outcome appears in all three events?' In this case, subtracting the pairwise intersections has now excluded these outcomes three times – once for each pair. Therefore, to adjust for this, we add back the probability of the triple intersection, which is the occurrence of all the events together. Through this balancing act, we can accurately calculate the chance of any one of the events happening by ensuring that each outcome is counted exactly once.

When applied to a problem like finding the probability of the union of three events, the Principle of Inclusion-Exclusion provides a formula that reliably accounts for overlapping occurrences. This principle can be extended to any number of events, making it versatile and powerful for various complex probability tasks.

Understanding probability of intersections is key to mastering the calculation of event unions. An intersection of events, denoted as, for instance, P(A ∩ B), refers to the probability that both events A and B happen simultaneously. It’s important to note that events can be independent, where the occurrence of one event doesn't affect the probability of the other, or dependent, where one event's occurrence does affect the other.

When calculating the probability of the union of events, it's essential to account for intersections correctly. For two events, this is fairly straightforward. However, when we add a third event, the intersections become more complex, involving pairwise intersections (like P(A ∩ B)) and the triple intersection (like P(A ∩ B ∩ C)).

This ties back to the Principle of Inclusion-Exclusion: after initially adding the probabilities of each event, we subtract the pairwise intersections to remove the overcounting. If there’s a chance all three events occur together, their intersection must be added back to the total probability to maintain accurate counting. This ensures we are including each possible outcome just once in our final calculation.

At the heart of these calculations is probability theory, which is a branch of mathematics that deals with the analysis of random events. The foundation of probability theory lays out the basic framework for assessing how likely it is for a particular event to occur. It provides the tools and principles necessary to evaluate risks and make sense of events that happen by chance.

Probability theory encompasses the rules that guide us in the calculation of probabilities for complex events. We use it to determine the likelihood of single events, like flipping a coin, but also to examine the relationships between multiple events, such as intersections and unions. The theory informs us that the total probability for all possible outcomes in a given situation must equal 1, setting a logical boundary for our calculations.

The way probability theory is applied in situations like finding the probability of the union of events showcases its importance in decision-making processes in a variety of fields - from finance to medicine, from engineering to social sciences. By understanding these core principles, students, professionals, and enthusiasts can not only solve textbook problems but also navigate real-world scenarios involving uncertainty and risk.