The Addition Rule for probabilities is a fundamental concept in probability theory that provides a way to calculate the probability of the union of two or more events. The rule applies to both mutually exclusive events (events that cannot occur simultaneously) and non-mutually exclusive events (events that can occur simultaneously).

There are two forms of the Addition Rule, depending on whether the events are mutually exclusive or not:

1. **For Mutually Exclusive Events:**
– If events A and B are mutually exclusive (they cannot both occur at the same time), the probability of the union of A and B (denoted as \(P(A \cup B)\)) is the sum of the probabilities of each individual event:
\[ P(A \cup B) = P(A) + P(B) \]

2. **For Non-Mutually Exclusive Events:**
– If events A and B are not mutually exclusive (they can occur simultaneously), the probability of the union of A and B is calculated as the sum of the probabilities of each event minus the probability of their intersection (the probability that both events occur):
\[ P(A \cup B) = P(A) + P(B) – P(A \cap B) \]

– The term \(P(A \cap B)\) represents the probability that both events A and B occur.

The Addition Rule can be extended to more than two events. For three events (A, B, and C), the rule becomes:
\[ P(A \cup B \cup C) = P(A) + P(B) + P(C) – P(A \cap B) – P(A \cap C) – P(B \cap C) + P(A \cap B \cap C) \]

In general, for any number of events, the formula involves alternating addition and subtraction of the probabilities of intersections of events.

It’s important to note that when events are mutually exclusive, the probability of their intersection (\(P(A \cap B)\)) is zero. In such cases, the Addition Rule for non-mutually exclusive events simplifies to the formula for mutually exclusive events.

The Addition Rule is a fundamental tool for calculating probabilities in various scenarios, and it is widely used in probability theory and statistics. It provides a systematic way to determine the likelihood of the occurrence of one or more events.