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ove to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 logic 2 probability 3 statistics 4 see also 5 notes 6 references toggle the table of contents mutual exclusivity 16 languages чӑвашла ελληνικά español فارسی 日本語 norsk bokmål português română русский slovenščina shqip ไทย українська اردو 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from mutually exclusive two propositions or events that cannot both be true this article is about logical exclusivity of events and propositions for the concept in concurrent computing see mutual exclusion for the concept in developmental psychology see mutual exclusivity psychology this article includes a list of general references but lacks sufficient corresponding inline citations please help improve this article by introducing more precise citations october 2009 learn how and when to remove this message part of a series on statistics probability theory probability axioms determinism system indeterminism randomness stochastic probability space sample space event collectively exhaustive events elementary event mutual exclusivity outcome singleton experiment bernoulli trial probability distribution bernoulli distribution binomial distribution exponential distribution normal distribution pareto distribution poisson distribution probability measure random variable bernoulli process continuous or discrete expected value variance markov chain observed value random walk stochastic process complementary event joint probability marginal probability conditional probability independence conditional independence law of total probability law of large numbers bayes theorem boole s inequality venn diagram tree diagram v t e in logic and probability theory two events or propositions are mutually exclusive or disjoint if they cannot both occur at the same time 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 so these two possibilities together exhaust all the possibilities 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 cannot happen at the same time but not collectively exhaustive there are other possible outcomes 2 3 5 6 logic edit see also logical possibility in logic two propositions ϕ displaystyle phi and ψ displaystyle psi are mutually exclusive if it is not logically possible for them to be true at the same time that is ϕ ψ displaystyle lnot phi land psi is a tautology to say that more than two propositions are mutually exclusive depending on the context means either 1 ϕ 1 ϕ 2 ϕ 1 ϕ 3 ϕ 2 ϕ 3 displaystyle lnot phi _ 1 land phi _ 2 land lnot phi _ 1 land phi _ 3 land lnot phi _ 2 land phi _ 3 is a tautology it is not logically possible for more than one proposition to be true or 2 ϕ 1 ϕ 2 ϕ 3 displaystyle lnot phi _ 1 land phi _ 2 land phi _ 3 is a tautology it is not logically possible for all propositions to be true at the same time the term pairwise mutually exclusive always means the former probability edit in probability theory events e 1 e 2 e n are said to be mutually exclusive if the occurrence of any one of them implies the non occurrence of the remaining n 1 events therefore two mutually exclusive events cannot both occur formally said x displaystyle x is a set of mutually exclusive events if and only if given any e i e j x displaystyle e_ i e_ j in x if e i e j displaystyle e_ i neq e_ j then e i e j displaystyle e_ i cap e_ j varnothing as a consequence mutually exclusive events have the property p a b 0 displaystyle p a cap b 0 2 for example in a standard 52 card deck with two colors it is impossible to draw a card that is both red and a club because clubs are always black if just one card is drawn from the deck either a red card heart 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 order to draw both a red card and a club the probability of doing so in two draws depends on whether the first card drawn was replaced before the second drawing since without replacement there is one fewer card after the first card was drawn the probabilities of the individual events red and club are multiplied rather than added the probability of drawing a red and a club in two drawings without replacement is then 26 52 13 51 2 676 2652 or 13 51 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 occurring 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 or a king drawing any 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 at least one of the events will occur is equal to one 4 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 one of flipping either a head or a tail events can be both mutually exclusive and collectively exhaustive 4 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 5 statistics edit in statistics and regression analysis an independent variable that can take on only two possible values is called a dummy variable for example it may take on the value 0 if an observation is of a white subject or 1 if the observation is of a black subject the two possible categories associated with the two possible values are mutually exclusive so that no observation falls into more than one category and the categories are exhaustive so that every observation falls into some category sometimes there are three 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 each dummy variable having two mutually exclusive and jointly exhaustive categories in this example one dummy variable called 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 1 if age is in the range 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 nonsensically imply that an observed subject is both under 18 and between 18 and 64 then the dummy variables can be included as independent explanatory variables in a regression the number of dummy variables is always one less than the number of categories with the two categories black and white there is a single dummy variable to distinguish them while with the three age categories two dummy variables are needed to distinguish them such qualitative data can also be used for dependent variables for example a researcher might want to predict whether someone gets arrested or not using family income or race as explanatory variables here the variable to be explained is a dummy variable that equals 0 if the observed subject does not get arrested and equals 1 if the subject does get arrested in such a situation ordinary least squares the basic regression technique is widely seen as inadequate instead probit regression or logistic regression is used further sometimes there are three or more categories for the dependent variable for example no charges charges and death sentences in this case the multinomial probit or multinomial logit technique is used see also edit contrariety dichotomy disjoint sets double bind event structure oxymoron synchronicity mece principle mutually exclusive and collectively exhaustive notes edit 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 distinct collectively exhaustive and mutually exclusive outcomes of an experiment intmath com mutually exclusive events interactive mathematics december 28 2008 1 2 stats probability rules 1 2 scott bierman a probability primer carleton college pages 3 4 non mutually exclusive outcomes cliffsnotes archived from the original on 2009 05 28 retrieved 2009 07 10 references edit 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 v t e philosophy branches applied philosophy logic metaphilosophy philosophical methodology philosophy of education philosophy of information philosophy of language philosophy of mathematics philosophy of religion philosophy of science political philosophy practical philosophy social philosophy theoretical philosophy aesthetics aesthetic response formalism institutionalism epistemology empiricism fideism naturalism particularism rationalism skepticism solipsism ethics consequentialism deontology morality absolutism nihilism particularism skepticism universalism value theory virtue metaphysics free will compatibilism determinism hard incompatibilism hard libertarianism mind behaviorism dualism eliminativism emergentism epiphenomenalism functionalism 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