Affinity Maturation (AM) is the process through which our immune system is able to develop potent antibodies against new pathogens it encounters, and is at the base of the efficacy of vaccines. At his core AM is analogous to a Darwinian evolutionary process, where B-cells mutate and are selected on the base of their affinity for an Antigen (Ag), and Ag availability tunes the selective pressure. When this selective pressure is high the number of B-cells quickly decreases and the population risks extinction in what is known as a population bottleneck. Here we study the probability for a B-cell lineage to survive this bottleneck as a function of the progenitor affinity for the Ag. Using recursive relations and probability generating functions we derive expressions for the average extinction time and progeny size for lineages that go extinct. We then extend our results to the full population, both in the absence and presence of competition for T-cell help, and quantify the population survival probability as a function of Ag concentration and population initial size.