Geometric interpretation of the Bernoulli-sum representation for the number of vertices

Determine a geometric interpretation of the fact that the number N_n of "true" vertices of the random convex chain T_n has the same distribution as a sum of independent Bernoulli random variables, where T_n is the convex hull of n i.i.d. uniform points in the triangle T with vertices (0,1), (0,0), and (1,0) together with the two vertices (0,1) and (1,0).

Background

Using recurrence relations for the probability generating functions G_n(t), the authors note that G_n has only negative real roots and hence N_n has the same distribution as a sum of independent Bernoulli random variables. While this facilitates probabilistic limit theorems, a direct geometric interpretation of this probabilistic decomposition is not known.

Clarifying the geometric mechanism behind this representation could link algebraic properties (orthogonality and root location of G_n) with the geometry of convex chains in the triangle model.