There is currently great interest in the development of methods for the minimally invasive diagnosis of fetal genetic disease using cell-free DNA from maternal plasma samples obtained in the first trimester of pregnancy. With the rapid development of high-throughput sequencing technology, the possibility of detecting the presence of trisomy fetal genomes in the maternal plasma DNA sample has recently been explored. The major concern of this whole genome sequencing approach is that, while detecting the karyotype of the fetal genome from the maternal plasma requires extremely high accuracy of copy number estimation, the majority of the available high-throughput sequencing technologies require polymerase chain reaction (PCR) and are subject to the substantial bias that is inherent to the PCR process. We introduce a novel and sophisticated statistical model for the whole genome sequencing data, and based on this model, develop a highly sensitive method of Minimally Invasive Karyotyping (MINK) for the diagnosis of the fetal genetic disease. Specifically we demonstrate, by applying our statistical method to ultra high-throughput whole sequencing data, that trisomy 21 can be detected in a minor (fetal) genome when it is mixed into a major (maternal) background genome at frequencies as low as 5%. This observation provides additional proof of concept and justification for the further development of this method towards its eventual clinical application. Here, we describe the statistical and experimental methods that illustrate this approach and discuss future directions for technical development and potential clinical applications.