Until recently, if a pregnant woman wanted to determine the genetic health of her baby, she had to endure a series of crude and invasive tests. Early in her second trimester, she would be given a blood test called the triple screen, which measures levels of three hormones in her bloodstream. The results of that test, combined with other factors such as the mother’s age, weight, and ethnicity as well as information gleaned from a high-definition ultrasound, provided a clue to whether her baby was at risk of having a chromosomal disorder (such as Down syndrome) or a neural tube...