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When can probabilities of separate events be added together?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

In probability theory, the probabilities of separate events can be added together when the events are mutually exclusive.12

Mutually exclusive events are those that have no outcomes in common, meaning the occurrence of one event precludes the occurrence of the others. Formally, if you have a set of events $A_{1}, A_{2}, \dots, A_{n}$ such that their intersection is the empty set ( $A_{i} \cap A_{j} = \emptyset$ for all $i \neq j$ ), then the probability of their union—the probability that at least one of these events occurs—is the sum of their individual probabilities.32

This principle of additivity is fundamental to the axiomatic theory of probability. It is important to distinguish this from the rule for independent events: when events are independent, the probability of their joint occurrence is calculated by multiplying, rather than adding, their individual probabilities.32