Mutually Exclusive Events
Two events are mutually exclusive (also called disjoint) when they cannot both occur on the same trial: they share no outcomes, so the probability of A and B together is zero. Rolling a die and getting a 2 and getting a 5 are mutually exclusive, as are a card being a heart and being a spade, or a customer's order being delivered and being cancelled in the same moment. The property concerns overlap in the sample space, the sets of outcomes having an empty intersection, and it says nothing about whether one event influences the other's likelihood, which is the separate property of independence treated in its own entry.
The practical payoff of mutual exclusivity is the simple addition rule: when events cannot overlap, the probability that at least one occurs is just the sum of their probabilities. For a fair die, P(roll is 1 or 2) = 1/6 + 1/6 = 1/3. The general rule, P(A or B) = P(A) + P(B) - P(A and B), reduces to this because the subtracted overlap term is zero. This is also Kolmogorov's third axiom of 1933, extended to any countable collection of pairwise disjoint events, which is why summing over disjoint cases is the engine behind counting arguments. If events are not mutually exclusive, adding their probabilities double-counts the overlap: the chance that a card is a heart or a king is 13/52 + 4/52 - 1/52 = 16/52, not 17/52, because the king of hearts belongs to both.
Mutually exclusive and independent are different properties, and in fact they are close to opposites: two events that can each happen are never both mutually exclusive and independent. Independence means P(A and B) = P(A) × P(B), so knowing B occurred does not change the chance of A. Mutual exclusivity means P(A and B) = 0. If both events have positive probability, the product P(A) × P(B) is positive, so it cannot equal zero, and therefore exclusive events with positive probability are always dependent: learning that a card is a heart drops the probability that it is a spade from 1/4 to exactly zero. The two ideas answer different questions (overlap in outcomes versus influence on probability) and the confusion between them is one of the most common errors in introductory probability, committed in both directions.
A mutually exclusive set of events that also covers the entire sample space is a partition, and partitions are the backbone of several central results. If the events B1, B2, ..., Bk are pairwise disjoint and together exhaust the sample space, their probabilities sum to exactly 1, and any event A can be decomposed along them: P(A) = sum of P(A and Bi). This law of total probability, combined with the conditional-probability definition, yields Bayes' rule, which is how a diagnostic test's performance is combined with a disease's prevalence, or a spam filter's word statistics with a base rate. A single event and its complement are the simplest partition. Mutually exclusive does not imply exhaustive: the events "roll a 1" and "roll a 2" are exclusive but leave other outcomes uncovered.
Mutual exclusivity is the structural assumption behind many modeling choices in data science. A multiclass classifier with a softmax output assumes each example belongs to exactly one class, so the class probabilities sum to 1; a multilabel classifier, in which an image may contain both a dog and a car, instead uses independent sigmoid outputs because the labels are not mutually exclusive. Categorical variables are encoded as one-hot vectors precisely because the categories are disjoint. In survey design, response options must be mutually exclusive and collectively exhaustive (the MECE principle used in consulting and questionnaire writing) so that every respondent fits exactly one bin. In reliability, competing failure modes are often treated as mutually exclusive events, since only the first failure ends the component's life.
Common pitfalls include assuming that events described as "different" are exclusive, and applying the simple addition rule when they are not. Everyday phrasing can mislead: "owns a car" and "commutes by bicycle" sound opposed yet overlap in many people, and "at least one of two machines fails" includes the case where both fail. The check is mechanical: ask whether any single outcome can satisfy both descriptions. Another error is concluding that because two events are exclusive, they must be unrelated; in reality, exclusivity is the strongest possible form of dependence, since one event's occurrence fully determines that the other did not occur. For sequential trials, events in different trials, such as heads on toss one and tails on toss two, are not exclusive at all, since both can happen and are usually independent.
Read mutually exclusive events as outcomes that cannot share a trial: it licenses adding probabilities without correction and slicing a sample space into clean cases, but it signals dependence rather than independence, since the occurrence of one event rules the other out.