Fetal DNA is present in the plasma of pregnant women. A fetus with trisomy of a chromosome will release an increased amount of DNA from that chromosome into maternal plasma. Such an increase has previously been measured using methods that allow individual DNA molecules to be counted. One such method involves the use of random massively parallel sequencing of maternal plasma DNA. As the sequencing process is random, sequence tags from a potentially aneuploid chromosome only represent a fraction of the sequencing data. The performance of selective amplification and sequencing of specific genomic regions is a recently reported approach for focusing the sequencing power onto genomic regions of diagnostic interest. This article provides a critical analysis of this approach and puts this method in the perspective of other recent works in the field.