In logic and probability theory , two propositions (or events) are **mutually exclusive** or **disjoint** if they are not both be true (occur). A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both.

In the coin-tossing example, both outcomes are, in theory, collectively exhaustive , which means that at least one of the outcomes must happen. ^{[1]} However, not all mutually exclusive events are collectively exhaustive. For example, the outcomes 1 and 4 of a single roll of a six-sided die are mutually exclusive (both can not happen at the same time) but not collectively exhaustive (there are other possible outcomes; 2,3,5,6).

## Logic

In logic , two mutually exclusive propositions are proposed that logically can not be true in the same sense at the same time. To be more than two propositions are mutually exclusive, depending on context, which may not be true, or at least one of which is true. The term *pairwise mutually exclusive* means that both can not be true simultaneously.

## Probability

In probability theory , events *E *_{1} , *E *_{2} , …, *E *_{n} are Said to be **exclusive Mutually** if the occurrence of Any One of Them Implies the nonoccurrence of the remaining *n* – 1 events. Therefore, two mutually exclusive events can not both occur. Formally said, the intersection of each of the two is empty: *A* ∩ *B* = ∅. In consequence, mutually exclusive events have the property: P ( *A* ∩ *B* ) = 0. ^{[2]}

For example, it is impossible to draw a card that is both red and a club because clubs are always black. If a card is drawn from the deck, a red card (gold or diamond) or a black card (club or spade) will be drawn. When *A* and *B*are mutually exclusive, P ( *A* ∪ *B* ) = P ( *A* ) + P ( *B* ). ^{[3]} To find the probability of drawing a red card or a club, for example, add together the probability of drawing a red card and the probability of drawing a club. In a standard 52-card deck, there are twenty-six red cards and thirteen clubs: 26/52 + 13/52 = 39/52 or 3/4.

One would have to draw at least two cards in a red card and a club. The probability of doing so in two draws is determined by the first drawing of the first drawing. The probabilities of the individual events (red, and club) are multiplied rather than added. 26/52 × 13/51 × 2 = 676/2652, or 13/51, the probability of drawing a red and a club in two drawings without replacement. With replacement, the probability would be 26/52 × 13/52 × 2 = 676/2704, or 13/52.

In probability theory, the word *or* allows for the possibility of both events happening. The probability of one or both events is denoted P ( *A* ∪ *B* ) and in general it equals P ( *A* ) + P ( *B* ) – P ( *A* ∩ *B* ). ^{[3]} Therefore, in the case of drawing a red card, a drawing of a red king, a red non-king, or a black king is considered a success. In a standard 52-card deck, there are twenty-six red cards and four kings, two of which are red, so the probability of drawing a red or a king is 26/52 + 4/52 – 2/52 = 28 / 52.

Events are collectively exhaustive if all the possibilities for outcomes are exhausted by those possible events, so at least one of those outcomes must occur. The probability that is at least one of the events will be equal to one. ^{[4]}For example, there are theoretically only two possibilities for coin flipping. Flipping a head and flipping a tail are collectively exhaustive events, and there is a probability of one of flipping either a head or a tail. Events can be both mutually exclusive and collectively exhaustive. ^{[4]}In the case of coin flipping, flipping and flipping are also mutually exclusive events. Both outcomes can not occur for a single trial (ie, 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. ^{[5]}

## Statistics

In statistics and regression analysis , an independent variable that can be taken over only two possible values is called a dummy variable. For example, it can take on the value 0 if an observation is of a small subject. The two possible categories are mutually exclusive, so that they are not exhaustive, so that each observation falls into one category. Sometimes there are two or more possible categories, which are pairwise mutually exclusive and are collectively exhaustive – for example, under 18 years of age, 18 to 64 years of age, and age 65 or above. In this case a set of dummy variables is constructed contents, each dummy variable HAVING two Mutually exclusive and exhaustive categories Jointly – in this example, one dummy variable (called Expired D _{1} ) Would equal 1 if age is less than 18, and Would equal 0*otherwise* ; a second dummy variable (called D _{2} ) would equal if 18-64, and 0 otherwise. In this set-up, the dummy variable pairs (D _{1} , D _{2} ) can have the values (1.0) (under 18), (0.1) (between 18 and 64), or (0.0) ( 65 or older) (but not (1,1), which would be nonsensically implied that they are both under 18 and between 18 and 64). Then the dummy variables can be included as independent (explanatory) variables in a regression. Note that the number of dummy variables is always variable, and there are two variable dummy variables in the two categories.

Such qualitative data can also be used for dependent variables . For example, a researcher might want to predict or not, using family income, a variable dummy gender, and so forth as explanatory variables. Here the variable is a variable dummy that equals 0 if the subject does not go to college and equals 1 if the subject does go to college. In such a situation, ordinary least squares (the basic technical regression) is widely seen as inadequate; rather probit regression or logistic regressionis used. Further, sometimes there are three or more categories for the variable variable – for example, no college, community college, and four-year college. In this case, the multinomial probit or multinomial logit technique is used.

## See also

- Dichotomy
- Disjoint sets
- Event structure
- Oxymoron
- Synchronicity
- Double bind

## Notes

**Jump up^**Miller, Scott; Childers, Donald (2012).*Probability and Random Processes*(Second ed.). Academic Press. p. 8. ISBN 978-0-12-386981-4 .The sample space is the collection or set of ‘all possible’ (collectively exhaustive and mutually exclusive) outcomes of an experiment.

**Jump up^**intmath.com; Mutually Exclusive Events. Interactive Mathematics. December 28, 2008.- ^ Jump up to:
^{a }^{b}Stats: Probability Rules. - ^ Jump up to:
^{a }^{b}Scott Bierman. A Probability Primer. Carleton College. Pages 3-4. **Jump up^**Non-Mutually Exclusive Outcomes. CliffsNotes.

## References

- Whitlock, Michael C .; Schluter, Dolph (2008).
*The Analysis of Biological Data*. Roberts and Co. ISBN 978-0-9815194-0-1 . - Lind, Douglas A .; Marchal, William G .; Wathen, Samuel A. (2003).
*Basic Statistics for Business & Economics*(4th ed.). Boston: McGraw-Hill. ISBN 0-07-247104-2 .