Suppose I have an urn with an infinite number of balls which can be either red or white. I do not know what the proportion of each colour is, but I do know it's a fixed proportion. After drawing $N$ balls, I have observed $r$ red ones and $w$ white ones.
I believe the probability that I will observe a red ball on the next draw is given by Laplace's Law of Succession, $\frac{r+1}{N+2}$. However, how sure should I be of that? That is, before I drew any balls, I believed any proportion other than $0$ or $1$ was the true one. After I drew those $N$ balls, what should be my estimate pdf over the possible values for the proportion of red balls in the urn?