Following an inequality of G. Pólya[1949], certain probability integrals, namelyof the limiting singular normal distribution of a specified multinomial, are shown to be bounded above by integrals Tk(Uk) of the uncorrelated standard multivariate normal density over k-dimensional hyperspheres *Ukof radius Uk=Uk(y) and center (0,…,0). The hypersphere Ukis so chosen that its k-dimensional volume is equal to the k-dimensional volume of a simplex Rk, the image under a diagonalizing transformation, of the simplicial region of integration fkin Φk(y,R). For a symmetric multinomial, Rkturns out to be a regular simplex and explicit formulas for the upper bound in the equicorrelated-equicoordinate case, with common correlation ρ= -1/k,- are derived in terms of (a) normal density and distribution functions and (b) incomplete gamma-function ratio.