An urn contains many balls of various colours. I sample without replacement, and I stop sampling whenever I draw the first Red ball. If it takes $n$ draws from an urn to draw the first Red ball, what is the proportion of Red balls in the urn?
Presumably, this fact gives some Probability Distribution Function over the proportion of Red balls in the urn. How do I figure out this PDF?
A commenter points out that it depends on my prior, as the update is done as follows (where $evidence$ is my observing a Red ball after $n$ draws:
$$P(proportion|evidence)=\frac{P(evidence|proportion)P(proportion)}{P(evidence)}$$
where $P(proportion)$ is my prior for the proportion of Red balls (of all balls in the urn). Suppose my prior is uniform on the $[0,0.01]$ interval. How do I calculate the PDF of the posterior proportion given my evidence?