Probability Definitions

Probability and Set Theory

Set: a collection of objects grouped together as one represented by curly brackets { }.
Element: an individual member or object of a set. It is represented by ∈. For example, s is a member or element of the set S:
s ∈ S.
Subset: a set containing a smaller number of elements from a set where all the elements of the subset belong to the set. It is indicated by ⊆. For example, a ⊆ S is read as: "a is a subset of S".
Empty set: a set that contains no elements. It is also called the null set or null. It is considered as a subset of all sets. It is represented by ∅ or { }.
Universal set: the set containing all the objects or sets having a certain property. It is represented by U.
Sample space: the total number or set of all possible outcomes. Also, it is called the probability space. It is represented by S. It can also be considered as the universal set.
Experiment: a procedure that can be repeated and has a defined result.
Outcome: the result of an experiment.
Event: the outcome of an experiment that is a subset of the sample space. Is represented by a noncapitalized letter. a ⊆ S is an event.

A Venn diagram is a graphical form of sets. The universal set or sample space is represented by a rectangle.

Subsets contained in the sample space or universal set are indicated by circles.

In this case A ⊆ U or A ⊆ S. A is a subset of U or A is a subset of S.

In the above diagram, B ⊆ A, B is a subset of A.

Also, B ⊆ A ⊆ U which is B is a subset of A which is a subset of U
and B ⊆ U where B is a subset of U

Union: a set containing all the elements of two or more sets being combined. It is represented by ∪.
Intersection: a set containing the elements of two or more sets that overlap or have the same elements in common. Is represented by ∩.

A ∪ B is read as "A union B". All the elements A are joined to the elements of B.

A ∩ B is read as "A intersect B".

The intersection contains only those elements in A that are also in B.

Complement: all the elements that do not belong in a set. It can be represented by A' or A^c. For example, if B is a set then B' or B^c would be the complement of B.

B^c is all the elements outside of B.