From an urn containing $r$ red and $w$ white balls, two balls are drawn one after the other without replacement. We imagine that the balls are labelled and so choos the model $\Omega=\{(k,l):1\leq k,l \leq r+w, k\neq l \}$ and $P=U_{\Omega}$ the uniform distribution. The labels $1,…,r$ stand for red and $r+1,…,r+w$ for white balls. We consider the event:
$$
B=\{\text{second ball is red} \}=\{ (k,l):l\leq r \}
$$
Why is $$|B|=r(r+w-1)$$ and $$|\Omega |=(r+w)(r+w-1)$$
Best Answer
$\lvert B \rvert$ counts the ways that after one ball (of either color) has been removed, the second ball is red. There are two ways that this can happen, and they are disjoint: the first ball is white, or the first ball is red. So, consider the disjoint events $E_1=\{\text{white then red}\}$ and $E_2=\{\text{red then red}\}$. Clearly $\lvert E_1\rvert = wr$ and $\lvert E_2\rvert = r(r-1)$. Then $\lvert B\rvert = \lvert E_1\rvert + \lvert E_2\rvert$ since these are disjoint, which produces the expression you have above if you squint.
For $\lvert\Omega\rvert$, this is "the number of choices for the first ball" times "the number of choices for the second ball". Let me know if that is still unclear.