In this paper we study a new probability associated with any given belief function b, i.e. the orthogonal projection π[b] of b onto the probability simplex P. We provide an interpretation of π[b] in terms of a redistribution process in which the mass of each focal element is equally distributed among its subsets, establishing an interesting analogy with the pignistic transformation. We prove that orthogonal projection commutes with convex combination just as the pignistic function does, unveiling a decomposition of π[b] as convex combination of basis pignistic functions. Finally we discuss the norm of the difference between orthogonal projection and pignistic function in the case study of a quaternary frame, as a first step towards a more comprehensive picture of their relation.