Union of Events and Addition Rule

The video introduces the concept of the union of events A and B, which includes outcomes belonging to A, B, or both. The notation for union is 'or' or the symbol '∪'.

The formula for the probability of the union of two events A and B is P(A∪B) = P(A) + P(B) - P(A∩B). The video explains that the intersection P(A∩B) is subtracted to avoid double-counting the outcomes present in both A and B, as illustrated by a Venn diagram.

The concept is extended to three events (A, B, C) with a more complex formula: P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C). An intuitive explanation with Venn diagrams demonstrates why each term is added or subtracted to ensure every region is counted exactly once.

A practical example is provided using a contingency table about a university proposal. The task is to find the probability that a randomly selected person is either a faculty member or in favor of the proposal. The solution applies the formula for two events, subtracting the intersection of faculty members who are in favor.

This example explores the possible range for the probability of the union of two events, A and B, given individual probabilities. It discusses two cases: when events are not mutually exclusive (where intersection is unknown) and when they are mutually exclusive (where intersection is zero). The answer is 'at most 77%' because if they are mutually exclusive, the sum is 77%, and if they are not, the sum is less due to subtracting the intersection.

Given probabilities for two events A and B, and their intersection, the video guides on constructing a Venn diagram. It’s revealed that the intersection is equal to the probability of A, implying that set A is entirely contained within set B. This peculiar arrangement clarifies the relationship between the sets.