91Ó°ÊÓ

The Inclusion-Exclusion Principle is a fundamental concept in probability theory. It helps calculate the probability of the union of multiple events. This principle corrects the mistake of simply adding probabilities of events, which can result in double-counting the probability of their intersection. The formula is:
\[\operatorname{Pr}(A \cup B) = \operatorname{Pr}(A) + \operatorname{Pr}(B) - \operatorname{Pr}(A \cap B)\]With it, you can calculate the combined probability of at least one of the events occurring.

• Used when events overlap.
• Prevents overestimation of probabilities.
• Can be extended to more than two events.

In our exercise, the principle is used to find the lower bound for the probability of both events occurring together (intersection). By rearranging the equation appropriately, it's shown that the probability of \(A \cap B\) must be at least 0.2.

The intersection of events in probability refers to scenarios where two or more events occur simultaneously. If you imagine an event called \(A\) and another event called \(B\), the intersection, shown as \(A \cap B\), is the probability that both events \(A\) and \(B\) happen at the same time.
In simple terms:

• \(A \cap B\) means "both A and B occur."
• It is a subset of each event.

In the exercise, we calculated that the minimum probability of \(A \cap B\) is 0.2 by using the inclusion-exclusion principle. This assures us that there is at least a 20% chance both events will happen together. Understanding intersections is crucial because it helps determine how events are related within a sample space.

In probability theory, the sample space \(\mathscr{S}\) is the set of all possible outcomes of a statistical experiment. It represents the "universe" or "context" in which probabilities are considered. If you think of rolling a die, the sample space is \(\{1, 2, 3, 4, 5, 6\}\) because these are all the possible outcomes.
Essential points about sample space:

• It includes every potential outcome.
• Serves as the foundation for assigning probabilities.
• The total probability of the sample space is always 1.

In our problem, the sample space \(\mathscr{S}\) is where events \(A\) and \(B\) reside. Understanding the sample space clarifies why \(\operatorname{Pr}(A \cup B) \leq 1\) in the step-by-step solution, reinforcing the concept that probabilities of overlapping events are considered within a set universe of possibilities.