Suppose, we flip a fair coin $k$-times. Now suppose $A_1, A_2, ... , A_{2^k}$ are $2^k$ distinct events, such that each of them has non-zero probability. Do there always exist such $2^k$ distinct events $B_1, B_2, ... B_{2^k}$, that satisfy the conditions:
1)$\forall i \leq 2^k \exists c_i \in \mathbb{N}$ such that $P(B_i) = \frac{1}{2^{c_i}}$
2)$\forall i, j \leq 2^k$ we have that $B_i$ and $B_j$ are independent iff $A_i$ and $A_j$ are independent.
I manually checked, that this is true for $k \leq 3$, but I do not know, whether this is true in general, or not.