The problem: Suppose you have 10 voters and 2 candidates. Each voter has a probability $p$ of voting for a Candidate 1, and a probability $1-p$ of voting for Candidate 2. What is the probability that exactly 6 voters will vote for Candidate 1, if $p$ is described by a Beta(3,3) function?
My thought process: I think I need to look for the marginal probability $P(X=6|n=10)$. I know that the beta function can be integrated as $30\int_0^1 x^2 (1-x)^2 dx$, but I'm not sure how I should go about solving this process. Could someone please get me started on the right path? Thank you!