Let's throw with a regular dice (probability)

Question

Let's throw with a regular dice twice (independently). Let $X_{1}$ be the number of dots thrown on the first try and let $X_{2}$ be on the second. We know, that $X=X_{1} + X_{2}$.

Calculate the expected value $\mathbb{E}[X_{1} \mid X = k ]$.

I'm stuck with this problem, help please.

Answer 1

Well, $k$ can be any resulting sum, so $2 \leq k \leq 12$.

Now if $k = 2$, what could $X_1$ be? Only 1, so $\mathbb{E}[X_1|X=2] = 1$.

Now if $k = 3$, what could $X_1$ be? Either we rolloed $(1,2)$ or $(2,1)$, so $X_1$ can be either $1$ or $2$ with equal probability, so $\mathbb{E}[X_1|X=3] = \frac{1}{2}1 + \frac{1}{2} 2 = 1.5$.

This you can do for any $k$ from 2 to 12...