91Ó°ÊÓ

The Inclusion-Exclusion Principle is a fundamental concept in probability and combinatorics. It allows us to calculate the probability of the union of multiple events. For two events, say \(A\) and \(B\), the principle states:

• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).

This formula helps us account for the overlap between \(A\) and \(B\), represented as \(P(A \cap B)\). If we simply added \(P(A)\) and \(P(B)\), we would count the overlap—the intersection—twice.
In the original exercise, this principle is adapted for conditional probability. We replace absolute probabilities with conditional probabilities, taking into account the specific conditions laid out by event \(C\). Thus, inclusion-exclusion becomes:

• \(P((A \cup B) \cap C) = P(A \cap C) + P(B \cap C) - P((A \cap B) \cap C)\).

This adjusted form accommodates our conditional perspective, ensuring we correctly calculate the effect of overlapping events concerning \(C\).

The union of events is a concept describing the scenario where at least one of the events occurs. In probability, when we refer to the union of two events \(A\) and \(B\), denoted as \(A \cup B\), we mean any outcome that belongs to either \(A\), \(B\), or both.
To mathematically determine the probability of the union of two events, we use:

• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

This is where the Inclusion-Exclusion Principle comes in handy to avoid counting duplicate outcomes. In conditional probability, applying this to events given \(C\) requires modifying each term to consider the condition, yielding:

• \(P(A \cup B | C) = \frac{P((A \cup B) \cap C)}{P(C)}\).

This expression shows how important it is to consider the context within which the events occur, especially when a condition, like \(C\), alters the original sample space from which probabilities are calculated.

In probability, the intersection of two events is fundamental in understanding how probabilities overlap. The intersection, denoted \(A \cap B\), includes outcomes common to both events \(A\) and \(B\).
The probability of the intersection \(P(A \cap B)\) helps find out how likely it is for both events to occur simultaneously. It's seen as the overlap that the Inclusion-Exclusion Principle adjusts for when calculating the union of events.
In a conditional probability framework, as examined in the exercise, we must think about intersections concerning the specified condition. Here, intersections involve conditional probabilities:

• \(P(A \cap B | C) = \frac{P((A \cap B) \cap C)}{P(C)}\)

This depicts the likelihood of both \(A\) and \(B\) happening, given \(C\) has occurred. This conditional approach is vital since it changes the whole sample space, focusing only on scenarios where \(C\) is true. Therefore, intersections in conditional probability require attentive consideration of how each event interacts under specific conditions.