The invention provides methods for determining the ploidy status of a chromosome in a gestating fetus from genotypic data measured from a mixed sample of DNA from the mother of the fetus and from thefetus, and from genotypic data optionally from the mother and from the father. The ploidy state is determined by using a joint distribution model to create a set of expected allele distributions for different possible fetal ploidy states given the parental genotypic data, and comparing the expected allelic distributions to the pattern of measured allelic distributions measured in the mixed sample,and choosing the ploidy state whose expected allelic distribution pattern most closely matches the observed allelic distribution pattern. The mixed sample of DNA may be preferentially enriched at a plurality of polymorphic loci in a way that minimizes the allelic bias, for example using massively multiplexed targeted PCR.