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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:

Ω = {h, t}

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:

ω ∈ Ω

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 Ω, then you will need a way to differentiate between the ω's. Some people use subscripts. For example:

ω1 ∈ Ω
ω2 ∈ Ω
ω3 ∈ Ω

...

ωn ∈ Ω

You could also write this more compactly as:

ω1 , ω2 , ω3 , ... , ωn ∈ Ω

In addition to often being denoted as ω, elements of Ω our 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:

Ω = {hh, ht, th, tt}

(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:

A ⊆ Ω

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

Continued Other Example

Continuing our example of the 2 coin flips, with:

Ω = {hh, ht, th, tt}

The event A that the first coin flip resulted in a heads is:

A = {hh, ht}

Also, the event B that the second coin flip resulted in a tails is:

B = {ht, tt}
-- Mirza Charles Iliya Krempeaux
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