What is $\omega$ in probability theory?

Question

I have a hard time understanding what is $\omega$ in probability theory. I understand that for a probability space $(\Omega, A, \mathbb{P})$, A is the sigma-algebra containing all the events which "may happen" ie are mesurable for $\mathbb{P}$. But when we consider $\omega \in \Omega$, what is it exactly ?

Answer 1

It is an element of $\Omega$--that is, a particular outcome.

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Added: To make the distinction between "outcome" and "event," let me give a particular example. Suppose our probability space were modeling the flipping of a fair coin twice. Then our outcomes (the results of the flips) are just ordered pairs with both components being either heads ($H$) or tails ($T$). That is, we want $$\Omega=\bigl\{\langle H,H\rangle,\langle H,T\rangle,\langle T,H\rangle,\langle T,T\rangle\bigr\}.$$ Now, we can make $\mathcal A$ the power set of $\Omega,$ and for any $A\subseteq\Omega$ (that is, for any $A\in\mathcal A$), let $\Bbb P(A)=\frac14|A|,$ where $|A|$ denotes the cardinality of $A$. Then $\langle\Omega,\mathcal A,\Bbb P\rangle$ is readily a probability space that models the situation precisely.

Now, one event that may transpire is that heads was flipped at least once. There are three possible outcomes such that this occurs--all but $\langle T,T\rangle$--so the event in question is $$\bigl\{\langle H,H\rangle,\langle H,T\rangle,\langle T,H\rangle\bigr\}.$$

Another event that may occur is that heads was flipped twice. There is only one outcome such that this occurs--namely $\langle H,H\rangle$--so the event in question is $$\bigl\{\langle H,H\rangle\bigr\}.$$

More generally, outcomes are elements of $\Omega,$ and elements of elements of $\mathcal A;$ events are subsets of $\Omega,$ and elements of $\mathcal A.$