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).
are mutually exclusive if it is not logically possible for them to be true at the same time; that is,
To say that more than two propositions are mutually exclusive, depending on the context, means either 1. "
is a tautology" (it is not logically possible for more than one proposition to be true) or 2. "
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.
As a consequence, mutually exclusive events have the property:
In a standard 52-card deck, there are twenty-six red cards and thirteen clubs: 26/52 + 13/52 = 39/52 or 3/4.
The probabilities of the individual events (red, and club) are multiplied rather than added.
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).
Events can be both mutually exclusive and collectively exhaustive.
Both outcomes cannot occur for a single trial (i.e., when a coin is flipped only once).
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 D1) would equal 1 if age is less than 18, and would equal 0 otherwise; a second dummy variable (called D2) would equal 1 if age is in the range 18–64, and 0 otherwise.
In this set-up, the dummy variable pairs (D1, D2) 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).
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.
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.