Poisson statistics & ratios

Question

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?

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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.

Answer 1

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.