Chapter 3.7, Problem 3.86ACB

Tests for Down syndrome. Currently, there are three diagnostic tests available for chromosome abnormalities in a developing fetus. The safest (to both the mother and fetus) and least expensive of the three is the ultrasound test. Two San Diego State University statisticians investigated the accuracy of using ultrasound to test for Down syndrome (Chance, Summer 2007).

Let D denote that the fetus has a genetic marker for Down syndrome and N denote that the ultrasound test is normal (i.e., no indication of chromosome abnormalities). Then, the statisticians desired the probability P (D / N). Use Bayes’s Rule and the following probabilities (provided in the article) to find the desired probability: P (D) = $\frac{1}{180}$ ,

P (D^c) = $\frac{79}{80}$, P (N|D) = $\frac{1}{2}$, P (N^c|D) = $\frac{1}{2}$, P (N|D^c) = 1, and P (N^c|D^c) = 0.