Is the statement $E[y - sign(s - 0.5)] = 0 $ with $s \sim Uniform(0,1) $ and $y \in \left\lbrace -1, 1\right\rbrace$ with $Prob(y=1)=s$ corect?

Question

Suppose $s \sim Uniform (0,1)$, $y \in \left\lbrace -1, 1 \right\rbrace$ with $Prob(y =1) = s$ and $Prob(y = -1) = 1-s$. One could think of $s$ as a signal telling how likely it is that $y$ will be $1$. I would like to check if \begin{equation} E \left[ y - sign(s - 0.5) \right] = 0 \end{equation} is a true statement. Does somebody have any idea for verifying the statement?

Answer 1

$$\mathbb{E}\left(y-\text{sign}\left(s-0.5\right)\right)=\mathbb{E}y-\mathbb{E}\text{sign}\left(s-0.5\right)=$$$$\mathbb{E}y-\frac{1}{2}\left(1+\left(-1\right)\right)=\mathbb{E}y=\mathbb{E}\mathbb{E}\left(y\mid s\right)=\mathbb{E}(2s-1)=0$$