Basics of Statistics and Linear Regression. Law of iterated expectations (LIE). Solving a problem.

Question

I am studying economics and I am confused by this problem Consider the three random variables (S, X, Y ) for which we have : Given data: X: 0 0 0 1 1 1 S: 1 2 3 1 2 3 Pr(X, S): 0.1 0.36 0.14 0.10 0.24 0.06 E(Y|X, S): 10 15 20 20 25 30

I need to Use the Law of Iterated Expectations to verify the following properties: a) E(Y ) = E[E(Y | X)] b) E(Y | X) = E[E(Y | S, X)| X] c) Let Z = f(S)=1 if S ≥ 2 0 else. Show that E(Y |Z) = E[E(Y | S)|Z]

I tried solving it but for a) I get 19 for the left hand side(LHS) and 9.44 for the RHS. I'm not sure if that is correct and I have no idea for the rest of them. Any help would be appreciated.