Two events are mutually exclusive if they cannot occur at the same time. An example
is tossing a coin once, 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. 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 but not collectively exhaustive.
Logic In logic, two mutually exclusive propositions
are propositions that logically cannot be true in the same sense at the same time. Another
term for mutually exclusive is "disjoint". To say that more than two propositions are
mutually exclusive, depending on context, means that one cannot be true if the other
one is true, or at least one of them cannot be true. The term pairwise mutually exclusive
always means two of them cannot be true simultaneously. Probability
In probability theory, events E1, E2, ..., En 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, the intersection of
each two of them is empty: A ∩ B = ∅. In consequence, mutually exclusive events have
the property: P(A ∩ B) = 0. 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 black card can be drawn. When A and B are mutually exclusive,
P(A ∪ B) = P(A) + P(B). 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 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 would depend on whether the first card drawn were
replaced before the second drawing, since without replacement there would be one fewer
card after the first card was drawn. The probabilities of the individual events would be multiplied
rather than added. The probability of drawing a red and a club in two drawings without replacement
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. 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). Therefore, if
one asks, "What is the probability 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. However, with mutually exclusive events the
last term in the formula, – P(A ∩ B), is zero, so the formula simplifies to the
one given in the previous paragraph. 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 1. 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. 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. 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. Statistics
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 male subject or 1 if the observation is of a female 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 would equal 1
if age is less than 18, and would equal 0 otherwise; a second dummy variable would equal
1 if age is in the range 18-64, and 0 otherwise. In this set-up, the dummy variable pairs can
have the values,, or, 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 variables
in a regression. Note that the number of dummy variables is always one less than the number
of categories: with the two categories male and female 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 goes
to college or not, using family income, a gender dummy variable, and so forth as explanatory
variables. Here the variable to be explained is a dummy variable that equals 0 if the observed
subject does not go to college and equals 1 if the subject does go to college. In such
a situation, ordinary least squares 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 college, community college, and four-year college.
In this case, the multinomial probit or multinomial logit technique is used.
See also Probability theory
Collectively exhaustive events Event structure
Disjoint sets Synchronicity
Dichotomy Notes
^ Miller, Scott , and Donald Childers. Probability and Random Processes. Academic Press, 2012.
p. 8: "The sample space is the collection or set of 'all possible' distinct outcomes
of an experiment." ^ Mutually Exclusive Events. Interactive Mathematics.
December 28, 2008. ^ a b Stats: Probability Rules.
^ a b Scott Bierman. A Probability Primer. Carleton College. Pages 3-4.
^ Non-Mutually Exclusive Outcomes. CliffsNotes. References
The Analysis of Biological Data, Michael C. Whitlock and Dolph Schluter.
Basic Statistics for Business & Economics, 4th edition, written by doctors Douglas A.
Lind, William G. Marchal, and Samuel A. Wathen.