I am doing some work on probability.

I have done some background reading on the definitions on exhaustive, non exhaustive and mutually exclusive but the definitions that I found do not make any sense to me.

Could someone please define exhaustive, non exhaustive and mutually exclusive and give a simple example of each definition.

I thank you in advance.

  • $\begingroup$ It is unclear what definitions were found to be unsatisfactory. Without knowing what those definitions were and what was wrong with them, it's hard to know what kind of improvement could be made. $\endgroup$ – David K May 23 '17 at 5:39

A collection of events is exhaustive if at least one of them must occur. A collection of events is non-exhaustive if it is possible for none of them to occur. Events are mutually exclusive if no two of them can occur simultaneously.

Let $X$ be the result of rolling one six-sided die, and define events $E_{\mbox{odd}}$ (the result of the roll is odd) and $E_i$ (the result of the roll is $i$).

The collection of events $\{E_{\mbox{odd}},E_1,E_2,E_4,E_6\}$ is exhaustive. (The result must be odd, $1$, $2$, $4$, or $6$.) The collection $\{E_1,E_2,E_3,E_4\}$ is not exhaustive (the result could be $5$ or $6$), though its events are mutually exclusive. The events in $\{E_{\mbox{odd}},E_2,E_4\}$ are mutually exclusive (no two can occur; also, the events are not exhaustive), and the events in $\mathcal E=\{E_{\mbox{odd}},E_1,E_2,E_3,E_4,E_5,E_6\}$ are not mutually exclusive (though $\mathcal E$ is exhaustive), because both $E_{\mbox{odd}}$ and $E_1$ can occur simultaneously.

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  • $\begingroup$ Hi steve, Thanks for a quick response. Upon my reading, i found out that an event is said to be exhaustive if all the possible elementary events under the experiment are covered by the event. My understanding of this statement is if we consider only 5 possible outcome for a die then it would be considered as a non-exhaustive event?? $\endgroup$ – fs2ly Apr 3 '16 at 20:08
  • $\begingroup$ You’re right, although I think these terms generally refer to a collection of events, not a single event. I guess you would say a single event is exhaustive if it equals the union of all the possible elementary events and that it is not if it is a proper subset of that union. $\endgroup$ – Steve Kass Apr 3 '16 at 20:11

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