Hi I have a question related to the treatment effect.
Recently, I am reading literatures on treatment effect and have a question.
In the literatures, we denote the counterfactual outcomes as $Y_1$ and $Y_0$ where $Y_1$ is for the treated and $Y_0$ is for the untreated. Then, the observed outcome is $Y=W\cdot Y_1+(1-W)\cdot Y_0$ where $W$ is the indicator of the treatment.
Here, my first question is whether or not $E(Y_1|W=1)$ and $E(Y|W=1)$ are different?
Second, I found an equation that is as follows: $$ \begin{aligned} E(Y|W=1)-E(Y|W=0) &= E(Y_1-Y_0)\\ &+\{E(Y_1|W=1)-E(Y_1|W=0)\}P(W=0)\\ &+\{E(Y_0|W=1)-E(Y_0|W=0)\}P(W=1) \end{aligned} $$ where P() is the probability function. But, I can't derive the equation.
Please, Help Me! Thank you for your time spent to read this question.