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When can the probabilities of two events be added together?

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probability and statistics
Swiss commemorative stamp of mathematician Jakob Bernoulli, issued 1994, displaying the formula and the graph for the law of large numbers, first proved by Bernoulli in 1713.

In probability theory, the probabilities of two or more events can be added together when the events are mutually exclusive (also known as disjoint).12

Two events are considered mutually exclusive if they have no outcomes in common, meaning the occurrence of one event precludes the occurrence of the other. Symbolically, if events $A$ and $B$ are mutually exclusive, their intersection is the empty set ( $A \cap B = \emptyset$ ), and the probability of their union is the sum of their individual probabilities:32

$P (A \cup B) = P (A) + P (B)$

This principle, known as the addition rule, extends to any number of events $A_{1}, A_{2}, \dots, A_{n}$ provided that no two events share any outcomes ( $A_{i} \cap A_{j} = \emptyset$ for all $i \neq j$ ). In such cases, the probability that at least one of these events occurs is equal to the sum of their individual probabilities.12