Collectively exhaustive events

In probability theory, 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, 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

$A \cup B = S$

where S is the sample space.

Compare this to the concept of a set of outcomes which are mutually exclusive, 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.

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