The following is an exercise from Pinsky and Karlin's An Introduction to Stochastic Modeling (4th edition):
Suppose that the outcome $X$ of a certain chance mechanism depends on a parameter p according to $Pr\left\{ X=1\right\}=p$ and $Pr\left\{ X=0\right\}=1-p$ , where $0\leq p\leq 1$. suppose that $p$ is chosen at random, uniformly distributed over the unit interval $\left[0,1\right]$, and then, that two independent outcomes $X_1$, and $X_2$ are observed. What is the unconditional correlation coefficient between $X_1$ and $X_2$?
note: Conditionally independent random variables may become dependent if they share a common parameter.
I already figured out that $X$ is uniformly distributed. Moreover $\mathbb{E}\left(X_{1}\mid P=p\right)=p=\mathbb{E}\left(X_{2}\mid P=p\right)$ and therefore $\mathbb{E}X_{1}=\mathbb{E}P=\mathbb{E}X_{2}$. What should be the next steps? Especially how do I get $\mathbb{E}X_{1}X_{2}$?
Hint: $$\begin{align}\mathsf P(X_1X_2=1\mid P=p)~&=~\mathsf P(X_1=1, X_2=1\mid P=p)\\[1ex] &=~\mathsf P(X_1=1\mid P=p)\,\mathsf P(X_2=1\mid P=p) \\[1ex]&=~ p^2\\[2ex]\therefore\quad \mathsf P(X_1X_2=1) ~&=~\end{align}$$