I am a guy (x) who knows four other guys (y,z,v,w) and five women (a,b,c,d,e). On a given day, we all speak to another person with probability $p$. On a given day, how many girls I either speak with directly or I spoke with someone (boy or girl) who also spoke to her?
This is what I have done. I spoke with 2.5 women in expectation and 2 guys. I do not know how to add the women those guys spoke with that I did not, and the women that the women I spoke with spoke with, adjusting for the fact that they could be the same.
Okay tried again: if women do not know each other, the answer is that I know one woman with probability $p$+ I do not know her $1-p$, so
$p+p(1-q)+p(1-q)^2+p(1-p)^3+p(1-p)^4$
yet this answer is not completely good because women know each other with probability $p$.
Let $A$ be the event that either you speak to $a$ or you speak to someone who speaks to $a$, and let $A'$ be the complement of $A$ (i.e., $A' = \text{not}\,A$).
Let $k$ be the number of people other than $a$ that you speak to.
Let $q=1-p$. \begin{align*} \text{Then}\;\;P(A') &=q\left(\sum_{k=0}^8\left(\binom{8}{k}p^kq^{8-k}\right)q^k\right)\\[4pt] &=q\left(\sum_{k=0}^8\binom{8}{k}p^kq^8\right)\\[4pt] &=q^9\!\left(\sum_{k=0}^8 \binom{8}{k}p^k\right)\\[4pt] &=q^9\left(1+p\right)^8\\[4pt] \end{align*} Noting that the contribution of $A$ to the required expected value is just $P(A)$, it follows that the required expected value is just $$5P(A) = 5(1-P(A'))=5\left(1-\left(q^9\left(1+p\right)^8\right)\right)$$
More generally, to allow for easy testing, suppose there are $m$ men (including you), and $w$ women.
Then letting $n=m+w$, the expected value would be $$w\left(1-\left(q^{n-1}\left(1+p\right)^{n-2}\right)\right)$$ Testing the above formula using $$m=2,\;w=1,\;p={\small{\frac{1}{2}}}$$ yields an expected value of ${\large{\frac{5}{8}}}$, which can easily be verified by direct calculation.