## Bernoulli scheme

We've got probability space (Ω,F,P) for Bernoulli scheme with n trials and with success probability equal to p. $A_k$ means exactly $k$ successes in $n$ trials. Prove that for any $B\in F$ and for any $k$, $P(B|A_k)$ doesn't depend on $p$. So I thought of writing it this way: $P(B|A_k)=\frac{P(A_k\cap B)}{P(A_k)}$ and $P(A_k)=$${n}\choose{k}$$p^kq^{n−k}$, but what next? How do we know how $B$ looks like? It's not, against my intuition, $j$ successes in m trials, because their intersection would be an empty set...