Deterministic algorithms for computational distance between distributions Computational distance between sequences of distributions $\{X_i\}_{i \in \mathbb{N}}$ and $\{Y_i\}_{i \in \mathbb{N}}$ can be defined as the maximum, over all probabilistic polynomial time algorithms $A$, of
$$ \left|\underset{x \sim X_{n}}{\mathsf{Pr}}(A(x) = 1) - \underset{x \sim Y_{n}}{\mathsf{Pr}}(A(x) = 1)\right|, $$
for each $n \in \mathbb{N}$. We can, very similarly, define computational indistinguishability to be when the computational distance is a negligible function of $n$. Here is my question: can we use a deterministic polynomial-time algorithm in place of a probabilistic one without loss of generality? Can we say that the maximum computational distance is achieved by a deterministic algorithm? Will this algorithm be non-uniform?
Hint:

 Every randomized algorithm $A(x)$ can be expressed as a deterministic algorithm $A(x;r)$ where $r$ is randomly chosen.  Now what can you say about the value of that absolute difference for different choices of $r$?  What would you like to be able to say about it?