We develop a simple model of genetic diversity in growing spherical cell clusters, where cell division is confined to the cluster surface, and genetic drift -- the discrete nature of cells -- can be very important. This kind of growth occurs in cells proliferating in soft agar, and can also serve as a simple model of avascular tumors. Local fluctuations in the genetic diversity play a key role, and can lead to the extinction even of selectively advantageous strains. We calculate the survival probability for both circular and spherical range expansions, and find that it is strongly influenced by inflation at the frontier.*Collaboration with Max Lavrentovich.