We defined a random variable in a probability space $(\Omega, E, P)$ as a map $X: \Omega \rightarrow \mathbb{R}$. Unfortunately, I somehow have the impression that this term "random variable is used differently.
For example in the context of a hypergeometric Distribution. The german Wikipedia page says that this is: The number of successes after taking out the first n balls or whatever we are referring to in the context of a hypergeometric Distribution. This would suggest that it is a number instead of a map, but this cannot be right.
In my opinion, the only possible way to define a random variable in this context is given by $X:\Omega \rightarrow \{0,1\}$, $X(\omega)=1,$ if your draw was a success and Zero otherwise. The Problem is. The english Wikipedia page writes at some Point $P(X=k),$ where k is the number of successes. I do not see how this could refer to a meaningful map $X:\Omega \rightarrow \mathbb{R}$, but rather to a map $X:P(\Omega) \rightarrow \mathbb{R}$, where P denotes(just in this case) the power set. Could anybody explain to me, where I am wrong?
Does nobody have an idea?
It is a number associated to a particular drawing, so it really is a map. Let's make a concrete example. Suppose we have an urn with 3 white balls and 2 black balls. Now, we draw 2 balls from the urn. A particular drawing could be $\omega_1=(W,B)$. The number of white balls is 1 so if we define X to be the number of white balls in a drawing of two balls, we have $X(\omega_1)=1$. On the other hand, if we had drawn $\omega_2=(B,B)$ we would have $X(\omega_2)=0$.
If we now wonder what the probability is that $X=1$, we compute $$P(X=1)=P((W,B) \text{ or } (B,W))= P(W,B)+P(B,W)=P(W)\times P(B|W)+P(B)\times P(W|B)= \frac{3}{5}\times\frac{2}{4}+\frac{2}{5}\times\frac{3}{4}$$
In which $P(W)$ and $P(B)$ are the probabilities to draw white and black on the first draw resp. And in which $P(B|W)$ is the probability to draw black the second time if the first draw was white and likewise for $P(W|B)$.