Estimate standard error of population proportion based on one sample

Question

There is a basket with unknown number of red balls ($N_r$) and green balls ($N_g$). I draw a random sample of $k$ balls without replacement and see that $P_r$ of the balls are red and $P_g$ of the balls are green. The expected ratio of the true populations in the basket $N_r/N_g$ should be $P_r/P_g$. Can I say something about the certainty (standard error) of this prediction without knowing the total number of balls $N$? If that is not possible, can I estimate the error in case $N$ is known? What about the case when $N$ is unknown but $N >> k$?