Consider two probability distributions $D$ and $U$, over $n$-bit strings, where $U$ is the uniform distribution. We are not given an explicit description of $D$: we are only given black-box access, ie, we are only given a sampling device that can sample from $D$. Consider a sample $z in {0, 1}^{n}$, taken from either $D$ or $U$. We want to know which one is the case, and to do that, we consider polynomial-time algorithms that use the sampling device.

Intuitively, it seems obvious that the best polynomial-time algorithm that distinguishes which distribution the sample $z$ came from must have gotten that very $z$ as a sample at least once when running the sampling device. How do I mathematically formalize this statement?

Also, does this same intuition hold if we are given a polynomial number of samples as input (taken from either $D$ or $U$) instead of just one and are also given access to a black-box sampler for $D$? That is, the best algorithm for distinguishability has to “see” all the input samples at least once while running the sampler, if it were to decide that the input samples come from the distribution $D$?