Mary and Neal play a game. Neal chooses some subsets from a set of size $n$. Each turn, Mary can choose a subset of any size, and Neal has to say whether the subset contains at least one subset that Neal chose. After a polynomial number (in $n$) of such exchanges, Mary has to guess Neal's smallest subset. If she does so, she wins. (If there are many smallest subsets, Mary wins by guessing any of them.)
Under optimal strategy, can Mary have a probability of at least $1/poly(n)$ of winning the game?
If Neal chooses only one subset, then Mary can start with the whole set and trimming it down one element at a time, as shown here. In each step she knows exactly whether the removed element belongs to Neal's subset or not. But if Neal chooses many subsets, it could be that Mary trims down "the wrong path", so that in the end she gets some subset that Neal chooses, but not the smallest one. And if Mary guesses randomly, there are $2^n$ subsets, so the chance of winning is only $1/2^n$.
No, Mary cannot have a $1/poly(n)$ probability of winning the game.
Suppose Neal chooses all subsets of size $n/2$, as well as one subset of size $n/2-1$. Whenever Mary asks about a subset of size at least $n/2$, the answer is always yes, so Mary can find out the set of size $n/2-1$ only if she asks exactly that subset. But there are $\binom{n}{n/2-1}$ subsets of this size, which is exponential in $n$.