Concept of conditional variance

[thtang]

I would like to confirm if conditional expectation and conditional variance is similiar to conditional probability.

conditional probability example

P(A|B) = P(A and B) / P(B)

then is it true that

E[A|B] = E[A and B] / E[B]

and

Var[A|B] = Var[A and B] / Var[B] or E[A^2 | B] - (E[A|B])^2

?
Thanks

[lee_onion]

I think E(A|B) = E(A and B) / P(B)
and Var(A|B) = V(A and B) / P(B) = E(A^2|B) - (E(A|B))^2;