The Sample Space
- The sample space, , is the set of all possible outcomes, .
- Subsets of are events.
- The empty set contains no elements.
Example – Tossing a coin
Toss a coin once:
Toss a coin twice:
Then event that the first toss is heads:
Set Operations – Complement, Union and Intersection
Given an event, , the complement of is , where:
The union of two sets A and B, is set of the events which are in either A, or in B or in both.
The intersection of two sets A and B, is set of the events which are in both A and B.
The difference set is the events in one set which are not in the other:
If every element of A is contained in B then A is a subset of B: or equivalently, .
If A is a finite set, then denotes the number of elements in A.
An indicator function can be defined:
Two events A and B are disjoint or mutually exclusive if (the empty set) – i.e. there are no events in both A and B).
More generally, are disjoint if whenever .
Example – intervals of the real line
The intervals are disjoint.
The intervals are not disjoint. For example, .
A partition of the sample space is a set of disjoint events such that .
Monotone increasing and monotone decreasing sequences
A sequence of events, is monotone increasing if . Here we define and write .
Similarly, a sequence of events, is monotone decreasing if . Here we define . Again we write