DNA replication in Xenopus laevis is extremely reliable, failing to complete before cell division no more than once in 10 000 times; yet replication origin sites are located and initiated stochastically. Using a model based on 1D theories of nucleation and growth and using concepts from extreme-value statistics, we derive the distribution of replication times given a particular initiation function. We show that the experimentally observed initiation strategy for Xenopus laevis meets the reliability constraint and is close to the one that requires the fewest resources of a cell.