In terms of Newtonian mechanics, the state of a rigid body, which uniquely determines the time evolution, is fully described by the position $\mathbb R^3$ and orientation, an element of $SO(3)$, linear momentum, an element of $\mathbb R^3$, and angular momentum, an element of $\mathfrak{so}(3)$, the Lie algebra of infinitesimal rotations. This is a 12-dimensional space.
Let's consider a state at the moment the tetrahedron is released in which it ends up balanced on one of its vertices. It is clear that there is not a 12-dimensional neighborhood of states that also end up on a vertex: translations would not change it, small changes in initial orientation would most probably make it fall, a change in the direction of the linear momentum would keep it ending up on its vertex, but a change in it's magnitude probably not (you would have to specify the problem more precisely, like in the initial orientation), and finally a small change in angular momentum would most of the time, at least in two of the dimensions, make it fall as well.
What this shows is that the space of initial conditions is probably 5-dimensional or so. It might have some components of higher dimension depending on the problem's specifics, but always strictly lower dimensional than the full space of initial conditions. So whatever continuous probably distribution you put on that space, it will have measure (hence probability) 0.
