In layman's terms, two events are mutually exclusive if they cannot occur at the same time (i.e., they have no common outcomes). The best example is tossing a coin, which can result in either heads or tails, but not both. Both outcomes can't happen simultaneously.
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In logic, two mutually exclusive propositions are propositions that logically cannot both be true. Another term for mutually exclusive is "disjoint" To say that more than two propositions are mutually exclusive may, depending on context mean that no two of them can both be true, or only that they cannot all be true. The term pairwise mutually exclusive always means no two of them can both be true.
In probability theory, events E_{1}, E_{2}, ..., E_{n} are said to be mutually exclusive if the occurrence of any one of them automatically implies the nonoccurrence of the remaining n − 1 events. Therefore, two mutually exclusive events cannot both occur. Mutually exclusive events have the property: P(A ∩ B) = 0.^{[1]} For example, one cannot draw a card that is both red and a club because clubs are always black. If one draws just one card from the deck, either a red card or a club can be drawn. When A and B are mutually exclusive, P(A or B) = P(A) + P(B).^{[2]} One might ask, "What is the probability of drawing a red card or a club?" This problem would be solved by adding together the probability of drawing a red card and the probability of drawing a club. In a standard 52card deck, there are twentysix red cards and thirteen clubs: 26/52 + 13/52 = 39/52 or 3/4.
One would have to draw at least two cards in order to draw both a red card and a club. The probability would depend on whether the first card were replaced. The probabilities would be multiplied rather than added. Without replacement, there would be one fewer card after the first card was drawn. The probability of drawing the two cards would be 26/52 * 13/51 = 338/2652, or 13/102. With replacement, the probability would be 26/52 * 13/52 = 338/2704, or 13/104.
When events are not mutually exclusive, the word "or" allows for the possibility of both events happening. If they are inclusive events (i.e., nonmutually exclusive events), P(A or B) = P(A) + P(B) – P(A and B).^{[2]} Therefore, if one asks, "What is the probability of drawing a red card or a king?" drawing a red king is considered a success. In a standard 52card deck, there are twentysix red cards and four kings, two of which are red: 26/52 + 4/52 – 2/52 = 28/52.
Events are collectively exhaustive if all the possibilities for outcomes are exhausted, and at least one of those outcomes must occur. The probability that at least one of the events will occur is equal to 1.^{[3]} For example, there are theoretically only two possibilities for flipping a coin. Flipping a head and flipping a tail are collectively exhaustive events, and there is a probability of 1 of flipping either a head or a tail. Events can be both mutually exclusive and collectively exhaustive.^{[3]} In the case of flipping a coin, flipping a head and flipping a tail are also mutually exclusive events. Both outcomes cannot occur for a single trial (i.e., when a coin is flipped only once). The probability of flipping a head and the probability of flipping a tail can be added to yield a probability of 1: 1/2 + 1/2 =1.^{[4]}
In statistics, each observation should be mutually exclusive in order for them to be properly differentiated and organized into separate categories, (such as male and female). Unlike in logic, however, of the two mutually exclusive observations, one does not necessarily have to be false. They both can be true, just not at the same time in the same category, (statistically speaking, a person cannot be both male and female, but two different people can be). In fact, the textbook definition of mutually exclusive from a statistics perspective is, "A property of a set of categories such that an individual or object is included in only one category."^{[5]} Another definition from the same source also says, "The occurrence of one event means that none of the other events can occur at the same time."^{[6]} Essentially, in statistics the concept of something being mutually exclusive serves to prevent it from being counted more than once in the overall tally and has less to do with it being true or false over something else (although it is always preferable to count only true data).
