Let $x_1, x_2,..., x_m$ be iid binomial random variables (each with a number of trials n and probability of success in each trial p). Define a list of binary indicator variables $y_1,y_2,...,y_m$ for at least $K$ successes i.e.,
$ y_i = \begin{cases} 1 & \text{if } x_i\geq K\\ 0 & \text{else} \end{cases}$
What is the variance of $z=\sum_1^m y_i$ ?
We have $Var\,z=m\,Var\,y_1$, $$Var\,y_1=P(x_1\ge K)(1-P(x_1\ge K))=P(x_1\ge K)P(x_1<K))=(1-F(K-1))F(K-1) =(1-I_{1-p}(n-K+1,K))I_{1-p}(n-K+1,K)$$ according to this, where $F$ is the cdf of the binomial distribution with parameters $n$ and $p$, and $I_\cdot(\cdot,\cdot)$ is the regularized incomplete beta function. Thus, $$Var\,z=m\,(1-I_{1-p}(n-K+1,K))I_{1-p}(n-K+1,K).$$
