Defining a probability space for a simple card experiment I'd like to proof-check my reasoning below.
Consider the following experiment: "draw cards from a deck (with $52$ cards), with replacements, until a king comes out, and then register how many draws were needed."
I want to give a probability space $(\Omega,{\cal A},\Bbb P)$ to model this experiment. This is supposed to be very simple, since random variables are mentioned only in the next chapter of the book.
Since the sample space $\Omega$ must contain all the possible results of the experiment, which are the number of draws, I'd take $\Omega=\{1,2,3,\ldots\}$, and since $\Omega$ is countable, we put ${\cal A}=\wp(\Omega)$ (that is, every event is a random event). 
I guess that $A_n$ = "the first king is drawn at the $n$th step" would be just  $A_n=\{n\}$, and it suffices to define  $\Bbb P$ at these atoms: $$\Bbb P(\{n\}) = \left(\frac{48}{52}\right)^{n-1}\frac{4}{52}=\frac{12^{n-1}}{13^n}. $$
I would like to know if I am missing anything subtle here. It seems that this can be studied using a sample space of sequences of cards and a certain random variable, but as I mentioned above, I'd like to avoid these tools for now and focus on the mathematical formalism of this model. Thanks.

The probability space you defined is indeed a probability space and is one model of the situation. You could also define it in a slightly different way:
Let $\Omega=\{\text{all permutations of the cards}\}$ and let $\mathcal F=p(\Omega)$. You have a measurable space $(\Omega, \mathcal F)$. Define a random variable $X: \Omega \to \Bbb R$ where $\Bbb R$ is endowed with the probability measure you said. Pullback that measure through $X$ to get a measure on $(\Omega, \mathcal F)$. So you have a measure on on the permutations of the cards.
Either way is fine but I think this way (in my opinion) might capture what's going on a little better.
Edit I didn't actually look at your measure and just assumed the calculation was right.
