Stochastic transition of cancer cells between drug-sensitive and drug-tolerant persister phenotypes has been proposed to play a key role in non-genetic resistance to therapy. Yet, we show here that cancer cells actually possess a highly stable inherited chance to persist (CTP) during therapy. This CTP is non-stochastic, determined pre-treatment and has a unimodal distribution ranging from 0 to almost 100%. Notably, CTP is drug specific. We found that differential serine/threonine phosphorylation of the insulin receptor substrate 1 (IRS1) protein determines the CTP of lung and of head and neck cancer cells under epidermal growth factor receptor inhibition, both in vitro and in vivo. Indeed, the first-in-class IRS1 inhibitor NT219 was highly synergistic with anti-epidermal growth factor receptor therapy across multiple in vitro and in vivo models. Elucidation of drug-specific mechanisms that determine the degree and stability of cellular CTP may establish a framework for the elimination of cancer persisters, using new rationally designed drug combinations.