Set Theory #
Probability is based around set theory, or the mathematics of sets. A set is a collection of objects, called elements. There is no order to the elements in a set, and no duplicates. Sets are denoted by capital letters, and written using curly braces.
A subset is a set whose elements are all in another set. A superset is a set that contains all the elements of another set. A proper subset is a subset that is not equal to the original set. A proper superset is a superset that is not equal to the original set. Two sets are equal if they are subsets of one another.
Some symbols for sets:
- \(\in\) means “is an element of”
- \(\notin\) means “is not an element of”
- \(\subseteq\) means “is a subset of”
- \(\superseteq\) means “is a superset of”
- \(\subset\) means “is a proper subset of”
- \(\superset\) means “is a proper superset of”
We can often use mathematical notation to describe sets. For example, \(\{x^2|x = 1, 2, 3, 4, 5\}\) means “the set containing the squares of 1, 2, 3, 4, and 5”, or \(\{1, 4, 9, 16, 25\}\) .
The null set is a set with nothing inside. It is denoted by \(\emptyset\) .
The universal set is the set containing all possible elements. It is denoted by \(S\) . Every set is a subset of the universal set.
The complement of a set is the set of all elements that are not in the original set. It is denoted by \(A^c\) .
The union of two sets is the set containing all elements in either set. It is denoted by \(A \cup B\) .
The intersection of two sets is the set containing all elements in both sets. It is denoted by \(A \cap B\) .
The difference of two sets is the set containing all elements in the first set that are not in the second set. It is denoted by \(A \setminus B\) .
Two sets are disjoint or mutually exclusive if they have no elements in common (that is, \(A \cap B = \emptyset\) ).
A collection of sets is collectively exhaustive if the union of all the sets is equal to the universal set (that is, \(\bigcup_{i=1}^n A_i = S\) ).
A collection of sets is a partition if it is both mutually exclusive and collectively exhaustive.
De Morgan’s law says that \((A \cup B)^c = A^c \cap B^c\) .
Applying Set Theory to Probability #
The basic model for studying probability is an experiment, which consists of a procedure and observations.
An outcome of an experiment is any possible observation of that experiment.
The sample space of an experiment is the finest-grain partition of all possible outcomes.
An event is a set of outcomes of an experiment.