I think the following proposition is likely to be true. I'd like to know a proof of it if any.
Proposition Let $A$ be an integral domain, $K$ its field of fractions. Let $P_1, ..., P_n$ be prime ideals of $A$. Let $S = (A - P_1)\cap\cdots\cap(A - P_n)$. If we regard $A_S$ and $A_{P_1}, \ldots, A_{P_n}$ as subrings of K, then $A_S = A_{P_1}\cap \cdots\cap A_{P_n}$.
EDIT I came up with a proof thanks to the Bill's hint. Let $\alpha \in A_{P_1}\cap \cdots\cap A_{P_n}$. Let $I$ = {$x \in A; x\alpha \in A$}. $I$ is an ideal of $A$. Since $I$ is not contained in any $P_i$, it is not contained in $P_1\cup\cdots\cup P_n$ by Proposition 1.11 of Atiyah-MacDonald. Hence $\alpha \in A_S$. Therefore $A_{P_1}\cap \cdots\cap A_{P_n} ⊂ A_S$. The other inculsion is obvious.