I consider d-dimensional random vectors \( Y_1,\ldots,Y_n \) that satisfy a mild general position assumption a.s. The hyperplanes $$ (Y_i - Y_j)^\perp (1 \le i < j \le n)$$ generate a conical tessellation of the Euclidean d-space, which is closely related to the Weyl chambers of type \( A_{n-1} \) . I will present a formula for the number of cones in this tessellation which holds almost surely. For a random cone chosen uniformly at random from this random tessellation, I will address expectations for a general series of geometric functionals. These include the face numbers, as well as the conical intrinsic volumes and the conical quermassintegrals. All these expectations turn out to be distribution-free. In a similar fashion, I will shortly discuss a conical tessellations which is closely related to the Weyl chambers of type \( B_n \). I will present analogous formulas the number of cones in this tessellation and the expectations of the same geometric functionals for the random cones obtained from this random tessellation. The main ingredient in the proofs is a connection between the number of faces of the tessellation and the number of faces of the Weyl chambers of the corresponding type that are intersected by a certain linear subspace in general position.