Venn Diagrams of Disjoint and Overlapping Events

The figures below provide a visual illustration of the Addition Rule. Remind yourself that $$\bigcup$$ means "OR" and $$\bigcap$$ means "AND".

For this venn diagram that represents overlapping events, we can see that the Probability of A or B equals the probability of of A plus the probability of B minus the probability of A and B. This venn diagram shows that the addition of the areas of the two circles alone will cause double counting of the overlapping area. This is the basic concept that underlies the addition rule.

When we have overlapping events, the Addition Rule states: $$P(A \bigcup B) = P(A) + P(B) - P(A \bigcap B)$$

The next Venn diagram shows two events that are disjoint, meaning the two events do not overlap. In this situation there is no space where $$P(A \bigcap B)$$ exists. Therefore it is not necessary to subtract it because it's equal to 0!

When we have disjoint events, the addition rule states: $$P(A \bigcup B) = P(A) + P(B)$$