expected value of failures before success in n experiments let $Y \sim B(15, p)$ be the number of successful experiments out of 15 and let $X$ be the number of failed experiments before the first success. I need to calculate $E[X|Y=1]$ and get a final answer without $p$.
I defined $W$ as the number of experiments until the first success so $W \sim G(p)$ hence $X=W-1$ so $E[X|Y=1] = E[W|Y=1 - 1] = E[W|Y=1] - 1$ but when I try to calculate $E[W|Y=1]$ I get
$$E[W|Y=1] = \sum_{i=1}^{15}ip(1-p)^{i-1}=\sum_{i=1}^{15}\binom{i}{1}p(1-p)^{i-1}$$ and I don't know how I can get a solution without p from this
$$\mathbb{E} (X|Y=1) = \sum\limits_{i = 0}^{14} i \cdot \mathbb{P} (X=i|Y=1) =  \sum\limits_{i = 0}^{14} i \cdot \frac{ \mathbb{P} ( (X=i) \cap (Y=1) )}{  \mathbb{P} (Y=1)  }\  $$
$$\mathbb{P} ( (X=i) \cap (Y=1) ) = p \cdot (1-p)^{14}, \ \forall i \in \{0,1, \dots, 14\}, \ \ \ \mathbb{P} (Y=1) = 15p\cdot(1-p)^{14}.$$
Therefore,
$$\mathbb{E} (X|Y=1) = \frac{ \sum\limits_{i=0}^{14} i}{15} = 7 $$
$$\mathbb{E} (X|Y=1) = \sum\limits_{i = 0}^{14} i \cdot \mathbb{P} (X=i|Y=1) =  \sum\limits_{i = 0}^{14} i \cdot \frac{ \mathbb{P} ( (X=i) \cap (Y=1) )}{  \mathbb{P} (Y=1)  }\  $$
$$\mathbb{P} ( (X=i) \cap (Y=1) ) = p \cdot (1-p)^{14}, \ \forall i \in \{0,1, \dots, 14\}, \ \ \ \mathbb{P} (Y=1) = 15p\cdot(1-p)^{14}.$$
Therefore,
$$\mathbb{E} (X|Y=1) = \frac{ \sum\limits_{i=0}^{14} i}{15} = 7 $$