A material property is often described by a proportionality factor between a solenoidal vector (electric or magnetic induction, electric current density) and an irrotational vector (field intensity); in an isotropic dielectric, for example,D=εE.Often one wishes to calculate the effective ε for a random mixture of two homogeneous materialsAandB, with constantsεAandεB,filling volume fractionspandq. There are exact formulas for the cases of boundaries all parallel or all perpendicular to the field, and an approximate formula for spheresAin a matrixBwithp≪1.For a statistically isotropic random mixture with arbitraryp, many approximations of unknown reliability have been proposed. Recently two rigorous results have been derived. (1) A series expansion inεA−εBrequires, besidesp, additional statistical information, dependent on the probabilitiesp12...n(n)of findingnpointsr1,r2,...,rnall simultaneously in materialA. (2) Given onlyεA,εB,andp1(1)(=p=1−q),we can state thatε*⩽ε⩽εwhereε*andε*are calculable and attainable. The present paper reviews these rigorous theories and presents closer bounds, based on additional information dependent onp12(2)andp123(3).