Given a sequence of orthonormal polynomials on R, {pn}n≥0, with pn of degree n, we define the discrete probability distribution Ψn(x)=(Ψn,1(x),…,Ψn,n(x)), with Ψn,j(x)=(∑j=0n−1pj2(x))−1pj−12(x), j=1,…,n. In this paper, we study the asymptotic behavior as n→∞ of the Shannon entropy S(Ψn(x))=−∑j=1nΨn,j(x)log(Ψn,j(x)), x∈(−1,1), when the orthogonality weight is (1−x)α(1+x)βh(x), α,β>−1, and where h is real, analytic, and positive on [−1,1]. We show that the limitlimn→∞(S(Ψn(x))−logn) exists for all x∈(−1,1), but its value depends on the rationality of arccos(x)/π. For the particular case of the Chebyshev polynomials of the first and second kinds, we compare our asymptotic result with the explicit formulas for S(Ψn(ζj(n))), where {ζj(n)} are the zeros of pn, obtained previously in [2].