The success of alien species on oceanic islands is considered to be one of the classic observed patterns in ecology. Explanations for this pattern are based on lower species richness on islands and the lower resistance of species-poor communities to invaders, but this argument needs re-examination. The important difference between islands and mainland is in the size of species pools, not in local species richness; invasibility of islands should therefore be addressed in terms of differences in species pools. Here I examine whether differences in species pools can affect invasibility in a lottery model with pools of identical native and exotic species. While in a neutral model with all species identical, invasibility does not depend on the species pool, a model with non-zero variation in population growth rates predicts higher invasibility of communities of smaller pools. This is because of species sampling; drawing species from larger pools increases the probability that an assemblage will include fast growing species. Such assemblages are more likely to exclude random invaders. This constitutes a mechanism through which smaller species pools (such as those of isolated islands) can directly underlie differences in invasibility.