Resistance to drugs is a fundamental problem in the treatment of many diseases. In this work we consider the problem of drug resistance in cancer, focusing on random genetic point mutations. A recent result obtained by Komarova is that for the case of a single drug treatment, the probability to have resistant mutants generated before the beginning of the treatment (and present, including their progeny, at some given time afterward) does not depend on the cancer turnover rate. This implies that the treatment success will not depend on such rate. In this paper we show that the number of such resistant mutants must depend on the turnover rate, which will also be the case for the success of the treatment.