### 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

#### Complement

Given an event, , the **complement** of is , where:

#### Union

The **union** of two sets A and B, is set of the events which are in either A, or in B or in both.

#### Intersection

The **intersection** of two sets A and B, is set of the events which are in both A and B.

#### Difference Set

The **difference set** is the events in one set which are not in the other:

#### Subsets

If every element of A is contained in B then A is a **subset** of B: or equivalently, .

#### Counting elements

If A is a finite set, then denotes the number of elements in A.

#### Indicator function

####
An indicator function can be defined:

### Disjoint events

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, .

### Partitions

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