The Probability of the Sample Space is $1$: $P(\Omega) = 1$. Let's say we denote the sample space as $\Omega$.
One of the axioms of the probability function is that the probability of something happening is equal to $1$. Ok, fine, this makes sense. But, notationally, this is represented as $P(\Omega) = 1$. This is saying that the probability of the sample space is equal to $1$. In my mind, it seems that there is a subtle difference between these two statements: One says that some event must occur and the other says that the entire sample space must occur. However, obviously, it is not necessary that the entire sample space occur, right? Rather, at least one outcome in the sample space must occur.
I must be thinking about this incorrectly, so I would appreciate it if people would please take the time to clarify this.
And the sample space is not a set of sets, right? It's simply a set of elements (outcomes/endpoints). So we cannot justify this by saying that the sample space represents the union of all its elements, since its elements are not sets and the union operator only operates on sets.
 A: Yes, you're thinking about it incorrectly.
Here's another example. Let's say you're rolling a fair die, and you're interested in the event $A=\{\text{rolling an even number}\}$. If the die shows $6$, does it qualify as event $A$ happenning? It certainly does, because $6$ is an even number. But from the formal point of view, here our sample space is $\Omega=\{1,2,3,4,5,6\}$, and this event is $A=\{2,4,6\}\subseteq\Omega$. Note that as the outcome of a single roll we only obtained one even number, not all even numbers. And yet, although we didn't get all elements of $A$, we don't have a problem saying that the event "we rolled an even number" has happened.
Without delving into all the technicalities (of measurable sets, $\sigma$-algebras, etc.) here's what these concepts mean.


*

*The sample space $\Omega$ consists of all possible individual outcomes.

*An event is by definition a subset of the sample space: $A\subseteq\Omega$. Some events can be singletons: $A=\{x\}$, where $x\in\Omega$ is one of elementary outcomes. But events do not have to be singletons, and quite often (mostly, I'd say) consist of many elements.

*The probability function $P$ is defined on events, i.e. on subsets, $A$ of the sample space $\Omega$, not on individual elements of $\Omega$. With a slight abuse of notation, people speak of probabilities of individual outcomes; but technically speaking it's not quite correct: when somebody mentions $P(x)$ for an element $x\in\Omega$, they actually mean $P(\{x\})$.

*For an event $A=\{x_1,x_2,\ldots\}\subseteq\Omega$, the probability $P(A)$, which we call the probability of the event $A$, means the probability that any one of the outcomes in $A$ has happened. And that's the only reasonable interpretation, since a single experiment can only yield a single outcome, and can't possibly yield many outcomes at once.
In other words, when we're interested in the probability $P(A)$ of an event $A$, it means that we will be happy whenever any one of the elements of $A$ happens; we can call them the desirable outcomes.
