Let $s_i$ be a Bernoulli random variable with $Pr(s_i=1)=p \forall i\in N=\lbrace 1, \dots, n \rbrace$. $s_i$ and $s_j$ are independent for all $j\neq i$. I would like to evaluate the following expectation.
$$E\left[ \frac{\sum_{j\in N}s_j}{\sum_{j\in N} (1-s_j)} \mid s_k=0\right] = E\left[ \frac{\sum_{j\in N\setminus k}s_j}{n - \sum_{j\in N\setminus k}s_j}\right]$$
I realise this is the ratio of two Binomial random variables. I have come across several results for such variables (e.g. https://en.wikipedia.org/wiki/Binomial_distribution#Ratio_of_two_binomial_distributions) , but none that seem to apply to this specific case.
I'm also interested in a generalisation of this problem, specifically when it is not necessarily the case that $p_i=p_j$ (for $j\neq i$).