Editing Collectively exhaustive events (section)

{{Short description|Set of events whose union covers the entire sample space}}
{{no footnotes|date=September 2017}}
{{Probability fundamentals}}

In [[probability theory]] and [[logic]], a [[Set (mathematics)|set]] of [[Event (probability theory)|events]] is '''jointly''' or '''collectively exhaustive''' if at least one of the events must occur.  For example, when rolling a [[dice|six-sided die]], the events 1, 2, 3, 4, 5, and 6 are collectively exhaustive, because they encompass the entire range of possible [[Outcome (probability)|outcomes]].

Another way to describe collectively exhaustive events is that their [[Union (set theory)|union]] must cover all the events within the entire sample space. For example, events A and B are said to be collectively exhaustive if
:<math>A  \cup B = S</math>

where S is the [[sample space]].

Compare this to the concept of a set of [[mutually exclusive events]].  In such a set no more than one event can occur at a given time. (In some forms of mutual exclusion only one event can ever occur.) The set of all possible die rolls is both mutually exclusive and collectively exhaustive (i.e., "[[MECE principle|MECE]]"). The events 1 and 6 are mutually exclusive but not collectively exhaustive.  The events "even" (2,4 or 6) and "not-6" (1,2,3,4, or 5) are also collectively exhaustive but not mutually exclusive. In some forms of mutual exclusion only one event can ever occur, whether collectively exhaustive or not. For example, tossing a particular biscuit for a group of several dogs cannot be repeated, no matter which dog snaps it up.

One example of an event that is both collectively exhaustive and mutually exclusive is tossing a coin. The outcome must be either heads or tails, or p (heads or tails) = 1, so the outcomes are collectively exhaustive. When heads occurs, tails can't occur, or p (heads and tails) = 0, so the outcomes are also mutually exclusive.

Another example of events being collectively exhaustive and mutually exclusive at same time are, event "even" (2,4 or 6) and event "odd" (1,3 or 5) in a [[Experiment (probability theory)|random experiment]] of rolling a [[six-sided die]]. These both events are mutually exclusive because even and odd outcome can never occur at same time. The [[Union (set theory)|union]] of both "even" and "odd" events give sample space of rolling the die, hence are collectively exhaustive.