What is meant by the probability distribution of a verifier?
A probability distribution is how the verifier's knowledge of the unknown value is summarised. This idea is usually easier to grasp for Bayesian statisticians than classical ones.
Suppose that the unknown value $x$ is just a $n$-bit string, a verifier might start out with no information about $x$ and thinks all $2^n$ possible values are equally likely: a uniform probability distribution on $2^n$ values. After an interaction with the prover, the verifier might say &quot;I now know that $x$ is not one of these $k$ values&quot; and their knowledge could be summarised as a uniform distribution on $2^n-k$ values. Or they might say &quot;I now know that $x$ has even Hamming weight&quot;, giving a uniform distribution on $2^{n-1}$. They might say that &quot;I am now 90% sure that $x$ has even Hamming weight&quot; and give a two-valued histogram distribution with probability $1.8/2^{n}$ on even Hamming weight values and $0.2/2^n$ on odd Hamming weight values. They might even say &quot;I think that interpreted as an integer $x$ is (truncated) Poisson distributed with mean $\mu$&quot;. All of these are different types of knowledge that the verifier might infer about $x$ based on their interactions with the prover.
The important thing about a zero-knowledge proof is that whatever the verifier infers from their interactions with the prover, they could generate the same information and inferences on their own.