# Sample Space

In statistics a **sample space** is the set of all possible outcomes.

## Example Sample Space

Here is one example. Image that you are going to flip a coin. The possible results of this are:

- the coin lands on heads,
- the coin lands on tails.

Thus, for this example, our **sample space** is made up of two items.
(Namely, #1 *the coin landing on heads* and #2 *the coin landing on tails*.

Note that in this example, I'm pretending that actual real possibilities that *could* happen in real life won't (for my example) because I want to keep the example simple.
Such as:

- the coin landing on its edge,
- a bird swooping down and catching the coin in its mouth and flying away with it,
- etc.

I'm also ignoring things, like rotational orientation that the coin lands with.
All I care about, in this example of mine,
is if *heads* is facing up for the coin when it lands,
or if *tails* is facing up for the coin when it lands.

## Denoting Sample Spaces

When people talk about **sample spaces**, they often use mathematical notation related to sets.
So, **sample spaces** are sets,
in the mathematical sense.

It is common for people to represent a **sample space** with the *capital* greek letter omega:
`Ω`

And thus our **sample space**, denoted by ** Ω**, is a set.

(Of course, you could represent a **sample space** with anything you want.
But if you are going to use something other than ** Ω**,
then it would probably be a good idea to make an effort to make it clear what you mean.)

## Continued Example Sample Spaces

Continuing the example from before, we could write that:

Where `h` denotes that the coin landed on heads,
and `t` denotes that the coin landed on tails.

## Elements, Points, Sample Outcomes, Realizations

When someone wants to talk about an element of a set ** Ω**, they often use the

*lowercase*greek letter omega:

`ω`So, using typical math notation, we could write:

Of course, this only works if you want to talk about a single element of the set ** Ω**.
If you want to talk about multiple elements of the set

**, then you will need a way to differentiate between the**

`Ω`**'s. Some people use subscripts. For example:**

`ω`...

You could also write this more compactly as:

In addition to often being denoted as ** ω**,

**elements**of

**our**

`Ω`**sample space**are also (sometimes) called:

**points**,

**sample outcomes**, and

**realizations**.

## Another Example

Here is another example. Image we are flipping a coin, again. But this time we are flipping it twice.

Thus, we can write that:

(Where `hh` means that the 1st coin flip gave you *heads* and the 2nd coin flip also gave you *heads*.
Where `ht` means that the 1st coin flip gave you *heads* but the 2nd coin flip gave you *tails*.
Where `th` means that the 1st coin flip gave you *tails* but the 2nd coin flip gave you *heads*.
Where `tt` means that the 1st coin flip gave you *tails* and the 2nd coin flip also gave you *tails*.)

## Events

A subset of our **sample space** (which we might denote as `Ω`)
is called an **event**.

There is no common convention for denoting **events** (like there are for **sample spaces**).
But, just for the sake of explaining this, let's say that we denote an **event** by ** A**,
then using typical math notation we could write:

Or, in other words, `A` is a subset of `Ω`.

## Continued Other Example

Continuing our example of the 2 coin flips, with:

The **event** ** A** that the first coin flip resulted in a

*heads*is:

Also, the **event** ** B** that the second coin flip resulted in a

*tails*is: