Finding the probability function Jug contains $8$ balls, $3$ are red and $5$ are blue.pulling out balls with no return intill the first blue ball comes out $\underline{\color{red}\bullet\times3 \,\,\,\,\,\color{blue}{\bullet}\times5}$ > Find the probability function of the number of balls that we need to pull out from the jug My try: Let $X$ be the number of balls that we need to pull out from the jug $$ \begin{array}{c|lcr} X&1&2&3&4 \\\ \hline \text{P}_\text{X}(x) & 5/8 & 5/7 &5/6 &5/5 \\\ \end{array} $$ But $\Sigma P_{X}(x)>1$

To obtain the first blue ball on draw $x$ you need to select $x-1$ of the three red balls, and then one of the five blue balls; out of all the ways to select $x$ of the eight balls. $$\mathsf P_X(x) = \dfrac{\dbinom{3}{x-1}\dbinom{5}{1}}{\dbinom{8}{x-1}\dbinom{9-x}{1}}\cdot\;\mathbf 1_{x\in\\{0,1,2,3,4\\}}$$