The law of total variance states that

Var[y] = E(Var(Y|X) + Var[E(Y|X)]

So:

Var(Y) = E(Var(Y|X)) if and only if Var(E(Y|X)) = 0

My question is when we have Var(Y) = E(Var(Y|X)) and therefore Var(E(Y|X)) = 0 **what does this say/imply about E(Y|X)** in this case?

Many thanks!