In exponentially proliferating populations of microbes, the population doubles at a rate distinct from than the average doubling time of a single-cell due to variability at the single-cell level. It is known that the distribution of generation times obtained from a single lineage is, in general, insufficient to determine a population’s growth rate. Is there an explicit relationship between observables obtained from a single lineage and the population growth rate? I will discuss two methods connecting the population’s growth rate to the statistics of single lineages. Both are related to a large deviation principle, that is a generic feature of exponentially proliferating populations. Intriguingly, the number of lineages needed to obtain an accurate estimate of the growth rate depends exponentially on the duration of the lineages, leading to a nonmonotonic convergence of the estimate, which we verify in both synthetic and experimental data sets.I will also discuss a connection of this inference problem to phase transitions in the Random Energy Model, an exactly solvable model of disordered systems where at low temperatures the partition function is contributed by rare, anomalously deep energy traps.