Tensor products and Residue fields

Question

Given a ring homomorphism between two Noetherian rings, $f:A \to B$. Let $P$ be a prime ideal in $B$ and let $\mathfrak{p}$ be an ideal in $A$ such that $f^{-1}(P) = \mathfrak{p}$. How can we prove the following:

$$B_P \otimes_A k(\mathfrak{p}) = B_P/\mathfrak{p}B_P$$

Answer 1

Let for instance $A$ be a field, and $P$ a prime ideal of $B$ (then $\mathfrak{p}=0$) such that $B\neq B_P$.