Consider a bipartite graph for which the two vertex sets have the same size, denoted $M=|V_1|=|V_2|$. Let $d$ denote the maximum degree of vertices in $V_1\cup V_2$. Assume that $M$ is large and that $d$ is small relative to $M$.
The experiment is to take a random sample of $10$ percent of $V_1$. Let $X$ be the portion of the vertices of $V_2$ that are connected to at least one member of the sample. I wish to prove that the variance of $X$ is small. More precisely, $\text{VAR}(X)\rightarrow 0$ in any sequence of such examples where $M\rightarrow\infty$ and $d/M\rightarrow 0$.
For example (in a simplified gender-binary world), when you sample $10$ percent of the females in the world, there should be small variance to the percentage of men in the world who are acquainted with at least one female from the sample.
In case it helps, I have the following formula for the expected value of $X$: $$E(X) \approx \sum_{d\geq 1} p(d)(1-.9^d),$$ where $p(d)$ is the portion of the vertices of $V_2$ that have degree $d$. This is because, for a vertex in $V_2$ of degree $d$, the chance that none of its $d$ edges connect to the sample is about $.9^d$.