I have three values ($x_1=2$, $x_2=8$, and $x_3=4$). These are drawn from a Binomial distribution with parameter $k=12$ and unknown parameter $p$. The prior distribution of $p$ is a Beta distribution with parameters $\alpha=3$ and $\beta=1$.
I know that the posterior would be a Beta distribution with parameters $\alpha+\sum x_i$ and $\beta+n-\sum x_i$.
So with that in mind, the answer would be $p \sim Beta(17,-10)$
But I'm confused on if:
1) I should be using the parameter $k$ somewhere or if it is not needed and
2) If a parameter can be negative or if I did something wrong.
I have very minimal knowledge of this subject and am just starting to learn.