# What is exactly $\omega$ $\in$ $\Omega$?

Example:

We toss a coin 3 times:

$$\Omega$$ = {$$\omega_1 \omega_2 \omega_3$$} = {HHH,HHT,HTH,TTT,THT,TTH,TTT}

2 times:

$$\Omega$$ = {$$\omega_1 \omega_2$$} = {HH,HT,TH,TT}

1 time:

$$\Omega$$ = {$$\omega_1$$} = {H,T}

What the heck does $$\omega_1$$ mean? I've read elsewhere that it's a singleton but really I don't know.

Does it mean:

{$$\omega_1 \omega_2$$} = {$$\omega_1 *\omega_2$$} = {H,T} * {H,T}

If we say that X is a stockprice that has on time $$X_1$$ three outcomes namely {Up(=U), Stays the same price(=EQ), Down(=D)} is

$$\Omega$$ = {$$\omega_1$$} = {U,EQ,D}

On time $$X_2$$

$$\Omega$$ = {$$\omega_1 \omega_2$$} = {U,EQ,D}*{U,EQ,D} = {UU,UEQ,UD,EQU,EQEQ,EQD,DU,DEQ,DDD}

I have looked at quite some answers like:

What is $\omega$ in probability theory?

Rigorous Meaning of "Drawing a Sample" $\omega$ from a Probability Space $(\Omega, \mathcal{A}, \mathbb{P})$

Probability Notation: What does $\{\omega\in \Omega : X(\omega) \in A\}$ mean?

But still I don't understand. Can someone please give an example with real numbers like a stock price or the temperature with if not's too much work a graph?

• It is worth emphasizing that there are many different ways to represent information, and many of the ways of representing information is not standard. It seems clear from context that $\omega_i$ somehow represents the outcome of the $i$'th coin flip or the $i$'th day of stock sales, etc... I would not have notated it as $\Omega = \{\omega_1\omega_2\}$ personally, but rather as $\Omega = \{\omega_1\omega_2~:~\omega_i\in\{H,T\}\}$ Mar 30, 2020 at 19:25
• So, when looking at one of the elements of $\Omega$ in the first question with the three coin flips, it might look like $HHT$ which indicates that the first flip was a head, the second flip also a head, and the third flip a tail. Mar 30, 2020 at 19:26
• I dislike this notation $\Omega = \{\omega_1\}$ and I would not use it personally. It is perfectly clear just to say $\Omega = \{ H, T \}$ instead. Here $\Omega$ is the set of all possible outcomes, and there are two possible outcomes. The notation $\{ \omega_1\}$ looks like a singleton, which I think is causing confusion. It would have been a little better, but still awkward, if they had said $\Omega = \{\omega_1 \mid \omega_1 = H \text{ or } \omega_1 = T \}$. Mar 31, 2020 at 3:55

What the heck does $$\omega_1$$ mean?
Here $$\omega_k$$ is being used to represent "the result of the $$k$$th coin toss".
As JMoravitz commented, your source is taking the notation $$\{\omega_1\omega_2\omega_3\}$$ to represent the outcome set generated by the first three coin toss results. That is, as an abbreviation for $$\{\langle\omega_1,\omega_2,\omega_3\rangle\in\{H,T\}^3\}$$
So, yes, that does mean $$\{\omega_1\omega_2\}$$ is the cartesian product $$\{\omega_1\}\times\{\omega_2\}$$
So when instead using your stockmarket exsample, with the options $$U,E,D$$. (Advise: don't use two letters when one will do, especially when concatenating them into strings.)
\begin{align}\{\omega_1\omega_2\}&=\{U,E,D\}^2\\&=\{UU,UE,UD,EU,EE,ED,DU,DE,DD\}\end{align}