The raw draw at a position: a float in [0, 1).
Whether an event of probability p happens at a position. Because the
underlying draw is fixed by the path, raising p only ever turns more
positions on.
An integer in the inclusive range [lo, hi] at a position.
A float in [lo, hi) at a position.
The samplers a generator needs, each addressed by the position it is drawing for rather than by how many draws came before it.
Every method is a pure function of the source and the path: asking twice gives the same answer, asking in a different order changes nothing, and a position nobody asks about costs nothing. That is what lets a caller change one parameter — or one bar — without redrawing the rest of the piece.