The probabilities of getting various values with dice can't be proved statistically. Statistical calculations assume things about these probabilities. The probabilities are what they are because of physics and symmetry.
IIRC there is a user on here named "seven-sided die." With all due respect to this user, the ... er ... typical dice we use are either regular polyhedra or shapes with almost as much symmetry as regular polyhedra. If you have such a die with no markings or etched numbers on it, then any face is equivalent to any other face, in the sense that you can rotate the die so that face A comes on top of face B, and when you have done so, the new state of the unmarked die is indistinguishable from the old state. The new surface coincides with the old surface, the center of mass is in the same place, and so on.
This symmetry implies that if you pick up the die without paying any attention to its orientation, and then throw it, then any initial throw you do is no more or less likely than any other throw in which the die has been reoriented to have B in place of A. Even if the way you throw is not random (e.g., you always throw your d6 from a certain orientation, with 5 on top), the physics of rolling dice is that the motion is exponentially sensitive to the initial conditions, so your choice doesn't actually influence the probabilities. There are a lot of processes that are random for this reason, including weather (on long time scales) or the fall of a pencil balanced on its tip.
This sensitivity to initial conditions could fail if the die doesn't really roll well, and that would be a reason not to use four-sided dice. (Or you could just make a habit of shaking them in your hand.) But the issue would be that a player could cheat, not that the die would intrinsically favor one side over the other if the player is honest.
Sagging Rufus wrote:
Because the D4 kind of just falls into positions, does that lower my chances of rolling X on generic dice?
If you pick up the d4 without looking at it or intentionally trying to orient it in a particular way, then by symmetry, for any motion you could trace out that leads to rolling 1, you would have some other motion that would lead to rolling 2. By symmetry, these initial conditions would be equally likely, and therefore the final results would be equally likely.