I am trying to make a connection between "naive" probability and Probability theory (in other words, undergraduate to graduate probability). That why of was thinking of the Binomial distribution (please correct me if I am wrong):
$\triangle$ "Naive" probabilistic approach.
take an experiment with two possible outcomes, success (:1) with probability $p$, failure (:0) with probability $1-p$. We repeat this experiment $n$ independent times. Set X to be the number of successes in these $n$ repetitions. So $X\sim Binom(n,p)$ and $\displaystyle P(X=k)=\binom{n}{k}p^k(1-p)^{n-k},~k=0,1,2,\ldots,n.$
$\Box$ Measuretheoretic approach.
1) The space of outcomes $\Omega$: it seems that this is fairly easy, it is $\{0,1\}^n$.
2) The $\sigma-$algebra $\mathcal{F}$: well, this is the $\sigma-$algebra generated by the class of all sets of the form $A_1\times \ldots \times A_n$, where $A_1,\ldots, A_n \subseteq\{0,1\}$ (QUESTION: what does the case $A_1=\emptyset$ represent in the real experiment described above?)
3) Probability measure $P$: This is $P(A_1\times \ldots \times A_n)=p(A_1)\ldots p(A_n)$, where $p(\{0\})=1-p,~p(\{1\})=p,$ extended on $\mathcal{F}$ according to Caratheodory's extension theorem.
4) Random variable $X$: We define $X:\Omega \rightarrow \mathbb{R}: X((\omega_1,\ldots,\omega_n))=\omega_1+\ldots+\omega_n =$ the number of successes. Now, $X$ is a random variable since for $k$ in $\{0,1,\ldots,n\}$ the set $\{X=k\}$ constists of all $\binom{n}{k}$ unions of the sets of the n-tuples of $\{0\}'$s and $\{1\}'$s, such that $\{1\}'$s appear $k$ times exactly. So, $\{X=k\}\in \mathcal{F}$ for all $k$.
5) Probability of the events of $X$: each n-tuple that appears in the above union has probability $p^k(1-p)^{n-k}$, by the definition of $P$ and since there are $\binom{n}{k}$ of them we get the known formula for $P(X=k).$
Is the setting correct?
Thanks a lot for any comments.
ps. It seems to me very interesting that the whole thing actually refers to the outcomes $\omega\in \Omega,$ instead of $X$ itself (although in an undergraduate probability course somebody might not even hear about a space $\Omega$ called the space of outcomes and all it's relatives).