The aggregate increased probability P of occurrence of any of n (presumed) nonthreshold end points caused by exposure to an environmental mixture of m toxic agents may be conveniently expressed under a few general assumptions. First, assume that the m agents are present in an environmental mixture at corresponding concentrations Ci, where i = 1,2,…,m, each of which produce, in exposed people, corresponding lifetime, time-weighted average biologically effective dose rates Dij, each causing one or more of n quantal (all or none) toxic end points Tj, where j = 1,2,…,n (see Figure I-1). Let Oij denote the occurrence of a particular jth end point Tj induced by effective dose rate Dij, and assume that Tj has a background occurrence probability of pj = Prob(Oij ¦ D=0) for total effective dose D due to all relevant agents and that Oij may arise only by events independent of those giving rise to either the background incidence rate of Tj or to events Ogh for any g and h such that g≠i, 1≤g≤m, h≠j, and 1≤h≤n. Finally, for very small values of Dij, assume that the corresponding increased probability of occurrence of the Tj is defined by an independent ''one-hit" (nonthreshold, low-dose linear) function of Dij. In the following, ∩, ∪, and the overbar denote the logical union, intersection, and negation operations, respectively.
It follows from the stated assumptions and definitions that a Dij-induced increased probability Pij of Tj occurrence, conditional on its independent background rate pj, is: