In probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, a set of events is jointly
or collectively exhaustive
if at least one of the events must occur. For example, when rolling a six-sided die
A die is a small throwable object with multiple resting positions, used for generating random numbers...
, the outcomes 1, 2, 3, 4, 5, and 6 are collectively exhaustive, because they encompass the entire range of possible outcomes.
Another way to describe collectively exhaustive events, is that their union must cover all the events within the entire sample space. For example, events A and B are said to be collectively exhaustive if
where S is the sample space.
Compare this to the concept of a set of outcomes which are mutually exclusive
In layman's terms, 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....
, which means that at most one of the events may occur. The set of all possible die rolls is both collectively exhaustive and mutually exclusive. The outcomes 1 and 6 are mutually exclusive but not collectively exhaustive. The outcomes "even" (2,4 or 6) and "not-6" (1,2,3,4, or 5) are collectively exhaustive but not mutually exclusive.
One example of a collectively exhaustive and mutually exclusive event is tossing a coin.
P(Head or Tail) = 1, so the outcomes are collectively exhaustive. When head occurs tail can't occur or P(Head and Tail) = 0, so the outcomes are mutually exclusive also.