How to obtain the Standard Deviation of a ratio of independent binomial random variables? X and Y are 2 independent binomial random variables with parameters (n,p) and (m,q) respectively.
(trials, probability parameter)

If $n$ and $m$ go to infinity while $p$ and $q$ are fixed, then the ratio $R=X/Y$ is well defined on an event of probability $1-o(1)$. 
On this event (or conditionally on this event, since the two asymptotically equivalent), Edgeworth expansions of $X$ and $Y$ show that the expectation of $R$ behaves like $\dfrac{np}{mq}$ and that its variance behaves like
$$
\left(\frac{np}{mq}\right)^2\left(\frac{1-p}{np}+\frac{1-q}{nq}\right).
$$
