My own intuition says that in probability theory the concept of random sampling is not contemplated. Randomness is used by probability theory but is not implied by its axioms. Randomly literally means that there is no law (expressed in equations) or algorithm (expressed in actions or some programing code) that connects one pick to another. The pick of set elements do not have to be obtained by subsequent action: they can be concurrent instead. Besides, they do not need to be independent, in the sense of probabilistic dependence. They can depend on each other while this dependence does not imply causation (in the sense that one implies the other).
I am supported in this by
Judea Pearl [1] when he stresses that probability is about "association" not "causality" (which is, in a sense, the reverse of randomness): An associational concept is any relationship that can be defined in terms of a joint distribution of observed variables, and a causal concept is any relationship that cannot be defined from the distribution alone. Examples of associational concepts are: correlation, regression, dependence, conditional independence, like-lihood, collapsibility, propensity score, risk ratio, odds ratio, marginalization, conditionalization, “controlling for,” and so on. Examples of causal concepts are:randomization, influence, effect, confounding, “holding constant,” disturbance, spurious correlation, faithfulness/stability, instrumental variables, intervention, explanation, attribution, and so on. The former can, while the latter cannot be defined in term of distribution functions." He also writes: " Every claim invoking causal concepts must rely onsome premises that invoke such concepts; it cannot be inferred from, or even defined in terms statistical associations alone."