Probability Theory Review
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Mutually Exclusive and Collectively Exhaustive Outcomes and Events
Consider the totality of possible outcomes that may result as a consequence of, or in fortuitous conjunction with, a particular activity. The set of all possible outcomes in a given problem situation is called the universal set and is typically denoted U. It is also known as the event space or outcome space in probability theory or sample space in statistics, where it is usually denoted S. We impose two conditions on this universe of possible outcomes:

1. The outcomes must be mutually exclusive. This means that only one outcome can occur at any one point in time. If an outcome occurs, it excludes the possibility of any other outcome occurring concurrently. Thus, when defining the universal set for a given problem, one must ensure there are no overlaps and ambiguities among the constituent outcomes.

2. The outcomes must be collectively exhaustive. This means that the universal set must include all eventualities relevant to the problem at hand. The implication is that one of the outcomes will inevitably occur in due course. Thus, when defining the universal set, one must ensure that all potential outcomes that could conceivably occur are included in U.

Since every universal set is a collection of mutually exclusive and collectively exhaustive outcomes, it follows that one and only one outcome will ultimately occur.

Events, however, contrary to outcomes, may or may not be mutually exclusive. This follows from the fact that events are collections of outcomes and that any given outcome can, in principle, be categorized as belonging to more than one event. (Recall that events are subjective categorizations defined by the problem analyst.)

Example 5: The outcome “2” in the roll of a die is an element of the event “Even” = {2, 4, 6} as well as the event “Less than 4” = {1, 2, 3}. These events are not mutually exclusive.

Example 6: The events “Even” and “Odd” are indeed mutually exclusive.

Events also need not be collectively exhaustive. Since events are defined according to their significance in the problem being analyzed and not all conceivable events may be deemed significant, a complete listing of possible events is generally not required.

Example 7: In the roll of a die, the events “Even” and “Less than 4” are not collectively exhaustive since the outcome “5” is unaccounted for.

Example 8: In the roll of a die, the events “Even” and “Odd” are collectively exhaustive.

Graphical Representation of Sets with Venn Diagrams
Venn diagrams provide an easy way of visualizing sets and their operations. The universal set of outcomes U is typically depicted as a rectangle. The points within the rectangle represent the possible outcomes associated with the activity of interest. The diagram is purely symbolic: the size of the rectangle U is irrelevant and the number of points it contains need not be construed as infinite (as is the case in geometry). Since U represents the totality of possible outcomes associated with a certain activity, we define its area to be 1 (meaning it contains 100% of the possibilities). Thus, U indicates that the set of possible outcomes is collectively exhaustive. And since its points do not overlap, they (and the outcomes those points represent) are taken to be disjoint or mutually exclusive.

An event E is usually depicted as an arbitrary circle within U. Again, size is irrelevant.

The above diagram can be used to portray the situation described in Example 1, where the activity was the roll of a die and E was the event “Even” = {2, 4, 6}. Note that E is a subset of U = {1, 2, … , 6}.  This is shown on the diagram by the fact that the circle E is wholly contained within U.

Complementary Events
In the above diagram, the remaining portion of U (the region not encompassed by E) depicts those outcomes that are not even numbers. This set of elements which are not members of E is called the complement of E or not E, and is denoted by the symbols E, E' or ¬E. Since E and E are mutually exclusive and collectively exhaustive events, their aggregation equals U. This is shown on the diagram by the fact that the area of E plus the area of E equals the area of U, meaning that the outcomes associated with event E = {2, 4, 6} plus those associated with event E = {1, 3, 5} equal the totality of outcomes in U.

An Event E and its complement E

Knowledge of the composition of U and of any event E implies full knowledge of E. Formally, E = U – E, the set difference between U and E.

 

Venn Diagrams

A Survey of Venn Diagrams