What if the distribution of a discrete random variable $X:\Omega \to \mathbb{R}$ is unknown. How do I calculate/show the probability of $X$ rigourously?
In the solutions of our problem sheets the professor only gives short explanations and then miraculously appears $P(X=k)$, the probability that $X$ attains $k$, where $P:X(\Omega) \to [0,1]$.
Is the following general intuition to prove $P(X=k)$ correct?
Count all $\omega \in \Omega$ with $X(\omega)=k$. Usually this includes some combinatorical arguments. Then figure out what $p(\omega)$ is. If you are lucky all $\omega$ are equally likely. If not then in most exercises we have an experiment which consists of several sub steps so that $p(\omega)= p_1(\omega_1) p_2(\omega_2)...p(\omega_n)$ or in case there are dependencies $p(\omega)= p_1(\omega_1) p_2(\omega_2|\omega_1)...p(\omega_n|\omega_1, ... \omega_{n-1})$. Finally, I sum up all $p(\omega)$ so that $P(X=k)\sum\limits_{\underset{\omega:~ X(\omega)=k}{\omega \in \Omega}}p(\omega)$.
Any comments are appreciated.