I'll start by motivating by question with a simpler scenario to ensure I've at least understood that scenario properly.
Scenario 1 :
Imagine an infinite sequence of numbers where $i$ is the $i^{th}$ element of that sequence. If I assign a random number $r_i \in (0,1)$ to each $i$, and then ask "What are the odds that for at least one $i$ we have $r_i = a$ ?" where $a$ is some specific pre-chosen number in $(0,1)$ (say $0.5$), I believe I am correct in assuming that such an $i$ will almost never exist because $\aleph_0 = |\mathbb{N}| < |(0,1)| = \mathfrak{c}$. Is this reasoning accurate ?
Scenario 2 :
So now I would like to consider another infinite set of random numbers, but this time I'd like to replace my discrete infinite sequence with a sort of continuous equivalent, which I suppose is somewhat similar to a random function. Namely, for every real number in $x \in (0,1)$ I associate a random number $r_x \in (0,1)$. Now I again ask the question "What are the odds that for some $x$ we have $r_x = a$ ?". Here I'm really not sure what to think. It's certainly possible that the very scenario I'm describing cannot be rigorously defined and no such set of $r_x$ can exist, in which case I'd appreciate an explanation as to why this is so and what the closest possible scenario is. Conversely, feel free to more rigorously define my problem.
Michael provided a solution to a seemingly related problem that's somewhere in between scenario 1 and 2 and so I'll post it here to see if anyone can extend this type of thinking to the reals :
1) Choose some natural number $n$
2) List the naturals from $1$ to $n$ (let's call this set $\mathbb{N}_n$)
3) For each number in $\mathbb{N}_n$ randomly assign another number in $\mathbb{N}_n$
We can then see that the probability that some randomly chosen number $a \in \mathbb{N}$ never appears is $(1-1/n)^n$, which approaches $1/e$ as $n\to\infty$. The chance that it does appear therefore approaches $1-1/e=0.632$.
Is there any way to extend this to all reals between $0$ and $1$ ?
After a little digging, I've come to think that perhaps there can't be an answer to my question, or if there is I'm probably in way over my head trying to find one. Let's define the problem a little more rigorously first :
1) Consider the family of sets $S_x$ where $S_x = (0,1) \forall x \in (0,1)$ (yes these are all the same set, why I'm considering this family will become apparent in step 2).
2) Using the axiom of choice, construct a set $R$ which contains one randomly chosen element from each $S_x$. This set $R$ is non-measurable.
3) Pick some number $a \in (0,1)$, and ask what are the odds that $a \in R$ is true ?
And now I'm simply not sure what to make of the fact that $R$ is non-measurable. Does it make my question unanswerable ? Or does it actually provide a great test case for understanding non-measurable sets ?
Here is a finite version of Scenario 1:
* List the rational numbers in $(0,1)$.
* Pick a large number $n$
* For each of the first $n$ numbers, associate one of the first $n$ numbers.
The probability that one of the first $n$ numbers never appears is $(1-1/n)^n$, which approaches $1/e$ as $n\to\infty$. The chance that it does appear approaches $1-1/e=0.632$