mutually exclusive (disjoint) events

Two events are mutually exclusive — also called disjoint — when they cannot both happen on the same run of the experiment. As sets, this means they share no outcomes at all: their intersection is empty, A ∩ B = the empty set. On a single die, 'the result is even' and 'the result is 1' are mutually exclusive, because 1 is not even, so no single roll can satisfy both. In a picture of two circles inside the sample space, mutually exclusive events are two circles that do not touch — there is no overlapping lens.

The reason this idea earns its own name is that it makes probabilities simple to add. When A and B are mutually exclusive, the chance that one or the other happens is just the sum of their separate chances: P(A or B) = P(A) + P(B), with no correction term. That clean rule is a special case of the general addition formula P(A or B) = P(A) + P(B) - P(A and B); since disjoint events have P(A and B) = 0, the subtraction vanishes. This additivity for non-overlapping events is so central it is built directly into the axioms of probability.

Two warnings prevent the usual mistakes. First, mutually exclusive is a fact about the sets, not about likelihood — events can be disjoint whether they are likely or rare. Second, and more important, mutually exclusive is not the same as independent; in fact they are nearly opposites. If two events with positive probability are mutually exclusive, then knowing one happened tells you the other definitely did not — that is a strong dependence, the very reverse of independence.