In probability, a set of events is **collectively exhaustive** if they cover all of the probability space: i.e., the probability of any one of them happening is 100%. If a set of statements is collectively exhaustive we know at least one of them is true.

These types of events or statements may or may not be mutually exclusive, since knowing that they cover all possibilities doesn’t tell us anything about whether or not they are redundant or whether two or may events may happen at the same time.

## Examples of Collectively Exhaustive Events

If you are rolling a six-sided die, the set of events {1, 2, 3, 4, 5, 6} is collectively exhaustive. Any roll must be represented by one of the set.

Sometimes a small change can make a set that is not collectively exhaustive into one that is. A random integer generated by a computer may be greater than or less than 5, but those are not collectively exhaustive options. Changing one option to “greater than or equal to five” or adding five as an option makes the set fit our criteria.

## Collectively Exhaustive Questions

In surveys it is important that multiple choice questions offer collectively exhaustive answer choices. For example, if a questionnaire asked if the respondent was:

- a. African
- b. Pacific Islander
- c. Latino
- d. Caucasian

an Asian respondent, for instance, would have no answer to mark.

This problem can be remedied if care is taken that every possible category is included, or, in cases where that is impractical, an ‘other’ option is included.

Since ‘other’ includes everything but previously listed choices, questions with an ‘other’ option added immediately meet this criteria.

## References

Baldwin (1914). “Laws of Thought”. Dictionary of Philosophy and Psychology. p. 23.

Kleene, Stephen C. (1952). Introduction to Metamathematics (6th edition 1971 ed.). Amsterdam, NY: North-Holland Publishing Company.

K., George. Mutually Exclusive & Coll. Exhaustive | Survey Tips. Researching & Marketing Strategies Blog. Published April 27th, 2010. Retrieved from https://rmsresults.com/2010/04/27/mutually-exclusive-collectively-exhaustive-survey-tips-market-research-syracuse-survey/ on August 16, 2018

Veneziano, Daniele. Brief Notes #1 Events and Their Probability. 1.151 Probability and Statistics in Engineering. Spring 2005. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA. Retrieved from https://ocw.mit.edu/courses/civil-and-environmental-engineering/1-151-probability-and-statistics-in-engineering-spring-2005/lecture-notes/briefnts1_events.pdf on August 16, 2018.