Events $A_1, A_2, A_3, \ldots$ are said to be mutually exclusive if the intersection of any pair of distinct events is empty, that is, $$(A_i \cap A_j) = \emptyset ~~\mathrm{for~all~} i\neq j.$$ Since the empty set has probability $0$, this implies that $P(A_i \cap A_j) = 0$. The third axiom of probability then tells us that $$P(A_1 \cup A_2\cup \cdots) = P(A_1) + P(A_2) + \cdots$$ and since $A_1 \cup A_2\cup \cdots \subset \Omega$, we have that the probability of the union cannot exceed $P(\Omega)=1$. Thus, $$P(A_1) + P(A_2) + \cdots \leq 1 ~\mathrm{for~mutually~exclusive~events~} A_1, A_2, A_3, \ldots$$ On the other hand, the collection of events $\{A_1, A_2, A_3, \ldots\}$ is said to be collectively exhaustive if $$A_1 \cup A_2\cup \cdots = \Omega,$$ that is, their union is the entire sample space. Neither of these properties implies the other. When a collection of events has both properties, it is said to be a partition of the sample space: we have partitioned (meaning divided up) the entire sample space into mutually exclusive events and so every outcome $\omega \in \Omega$ is a member of exactly one event in the partition. Example: If $\Omega = \{1,2,3,4\}$, then
Probability > 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 EventsIf 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 QuestionsIn surveys it is important that multiple choice questions offer collectively exhaustive answer choices. For example, if a questionnaire asked if the respondent was:
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. ReferencesBaldwin (1914). “Laws of Thought”. Dictionary of Philosophy and Psychology. p. 23. 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.
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In probability theory and logic, 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 events 1, 2, 3, 4, 5, and 6 balls of a single outcome 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 ∪ B = S {\displaystyle A\cup B=S} 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"). 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. HistoryThe term "exhaustive" has been used in the literature since at least 1914. Here are a few examples: The following appears as a footnote on page 23 of Couturat's text, The Algebra of Logic (1914):[1] "As Mrs. LADD·FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought"[2]), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. LADD-FRANKLIN proposes to call them respectively the principle of exclusion and the principle of exhaustion, inasmuch as, according to the first, two contradictory terms are exclusive (the one of the other); and, according to the second, they are exhaustive (of the universe of discourse)." (italics added for emphasis)In Stephen Kleene's discussion of cardinal numbers, in Introduction to Metamathematics (1952), he uses the term "mutually exclusive" together with "exhaustive":[3] "Hence, for any two cardinals M and N, the three relationships M < N, M = N and M > N are 'mutually exclusive', i.e. not more than one of them can hold. ¶ It does not appear till an advanced stage of the theory . . . whether they are 'exhaustive' , i.e. whether at least one of the three must hold". (italics added for emphasis, Kleene 1952:11; original has double bars over the symbols M and N).See also
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If two events are mutually exclusive and collectively exhaustive
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