Find $E(X_1X_2 \mid X_2 X_3)$ for i.i.d. symmetric Bernoulli random variables $X_k$

Question

I have problems with computing some basic conditional expected value - this is most likely done via transformations of CEV, but I can't get them done properly. Can anyone help me with the following task?

Let $X_1, \ X_2, \ X_3$ be iid random variables for which $P(X_i = \pm 1)=\frac{1}{2}$. Find $E(X_1X_2 \mid X_2 X_3)$.

Answer 1

Without brute force: $$ \begin{aligned} E[X_1X_2|X_2X_3] &= E[E[X_1X_2|X_2X_3,X_2]|X_2X_3] \\ &= E[X_2E[X_1|X_2X_3,X_2]|X_2X_3] \\ &= E[X_2E[X_1]|X_2X_3]\\ &= E[X_2\cdot 0 |X_2X_3]\\ &=0 \end{aligned} $$

First equation is the tower property for conditional expectations, second equation follows because $X_2$ is measurable with regards to the sigma algebra generated by $(X_2, X_2X_3)$ and the third one by independence of $X_1$ and $(X_2, X_2X_3)$.