# Poisson statistics & ratios

The rule of thumb I know for the error on a count for random sampling is that it goes as the square root of the count. For instance, if I observe a radioactive source for $$1$$ minute and measure $$100$$ decay events, a reasonable uncertainty on the count is $$100\pm\sqrt{100} = 100 \pm 10$$.

I have a different sampling problem, though. To put it simply, imagine I have a bag filled with a large number of balls which are each either red or blue. I'm interested in measuring the fraction of red balls in the bag. I draw an unbiased sample of $$N_{\rm balls}$$ balls and count the red ones, finding $$N_{\rm red}$$. My estimate for the red fraction $$f_{\rm red}=N_{\rm red}/N_{\rm balls}$$, but what uncertainty should I associate with this? I'm tempted to say $$f_{\rm red}=\frac{N_{\rm red}}{N_{\rm balls}}\pm\frac{\sqrt{N_{\rm red}}}{N_{\rm balls}}$$, but I find this unsatisfying. This is because it makes no distinction in the case where I draw $$N_{\rm red}=0$$; if this occurs with $$N_{\rm balls}=1$$, then my measurement is not especially constraining and should have a large uncertainty, but if I found $$N_{\rm red}=0$$ with $$N_{\rm balls}=10^5$$ I would have a high confidence that $$f_{\rm red}\sim0$$. In the particular experiment I have in mind, repeated draws are not an option.

Could someone point me to a correct handling of the statistics for such an experiment?

I appreciate that the physics content here is fairly minimal, so if the community prefers that this move to CV.SE or M.SE that's fair enough; I put it here because the context is a physical experiment and I would prefer answers in terms familiar to a physicist.

There is a problem here, in my opinion. You never can justify the guess $$f=0$$ rigorously it will always remain a hypothesis. If the bag contains an infinite number of balls and the fraction of red balls is $$f\neq 0$$, then a distribution of $$N_{\mbox{red}}$$ is the binomial distribution with the mean value $$\overline{N_{\mbox{red}}} = N_{\mbox{balls}} f$$ and the variance $$D = N_{\mbox{balls}} f (1-f).$$ In this case, $$N_{\mbox{red}}/N_{\mbox{balls}}$$ is near $$f$$ with hight probability only if $$\overline{N_{\mbox{red}}} \gg \sqrt{D}.$$ This inequality is equivalent to the following one: $$N_{\mbox{balls}} \gg \frac1{f}-1.\qquad(*)$$ So if $$f \ll 1$$, then $$N_{\mbox{balls}}$$ must be large enough to get a reliable estimation for $$f$$. If condition $$(*)$$ is not satisfied, then the probability of $$N_{\mbox{red}} = 0$$ is not small.