Phenotypic plasticity is an evolutionary driving force in diverse biological processes, including the adaptive immune system, the development of neoplasms, and the persistence of pathogens despite drug pressure. It is essential, therefore, to understand the evolutionary advantage of an allele that confers on cells the ability to express a range of phenotypes. Here, we study the fate of a new mutation that allows the expression of multiple phenotypic states, introduced into a finite population of individuals that can express only a single phenotype. We show that the advantage of such a mutation depends on the degree of phenotypic heritability between generations, called phenotypic memory. We analyze the fixation probability of the phenotypically plastic allele as a function of phenotypic memory, the variance of expressible phenotypes, the rate of environmental changes, and the population size. We find that the fate of a phenotypically plastic allele depends fundamentally on the environmental regime. In constant environments, plastic alleles are advantageous and their fixation probability increases with the degree of phenotypic memory. In periodically fluctuating environments, by contrast, there is an optimum phenotypic memory that maximizes the probability of the plastic allele's fixation. This same optimum memory also maximizes geometric mean fitness, in steady state. We interpret these results in the context of previous studies in an infinite-population framework. We also discuss the implications of our results for the design of therapies that can overcome persistence and, indirectly, drug resistance.Perpetual volatility in the surrounding microenvironment, nutrient availability, temperature, immune surveillance, antibiotics or other drugs are realities of life as a microorganism 1-3 . To persist in constantly changing environments and increase resilience, microbial populations often employ mechanisms that expand the range of phenotypes that can be expressed by a given genotype 3, 4 . This form of bet-hedging may not confer an immediate fitness benefit to any one individual, but it can sometimes act to increase the long-term survival and growth of an entire lineage 5,6 .Phenotypic heterogeneity has been well documented both in the context of cellular noise without any known environmental triggers 7 and in the context of persistent environmental challenges 8 . Classic examples include the bifurcation of a genotypically monomorphic population into two phenotypically distinct bistable subpopulations 1 , or phase variation, a reversible switch between different phenotypic states driven by differences in gene expression 3,4,9 . In order to motivate our modeling study, we first describe and compare four clinically relevant examples of phenotypic bet-hedging.One of the most striking examples of evolutionary bet-hedging is bacterial persistence 6, 10-12 , whereby a genetically monomorphic bacterial population survives periods of large antibiotic concentrations by producing phenotypically heterogeneous sub-populations, ...