When it first arises, a particular mutation will be on a unique haplotypic background. Assuming that the mutation arose only once, chromosomes in the current population which carry it will all share a region, IBD, around the mutation. We present a theoretical analysis, based on population genetics modelling and simulation, of the likely size of this shared region and of the way in which it depends on the population frequency of the mutation, and the demographic history of the population. These results are contrasted with those for regions shared by chance in arbitrary samples from the population. We discuss consequences for patterns of linkage disequilibrium and the required density of proposed SNP maps, and for their use in the design and analysis of genome-wide, association-based, studies for genes involved in common, complex, disorders. Our work exploits recent results on the structure of conditioned coalescents processes which describe the genealogical history of a sample of chromosomes under the assumption that a particular subset of the chromosomes carry a mutation assumed to have arisen uniquely.