Question

Tests for Down syndrome.Currently, there are three diagnostictests 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 Ddenote that the fetus has a genetic marker for Down syndrome and Ndenote that the ultrasound test is normal (i.e., no indication of chromosome abnormalities). Then, the statisticians desired the probability P(D/N). Use Bayes’sRule and the following probabilities (provided in the article) to find the desired probability:

$\mathit{P}\mathbf{\left(}\mathit{D}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{180}}\mathbf{,}\mathit{P}\mathbf{\left(}{\mathit{D}}^{\mathbf{C}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{79}}{\mathbf{80}}\mathbf{,}\mathit{P}\left(\frac{N}{D}\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\mathit{P}\left(\frac{N^C}{D}\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\mathit{P}\left(\frac{N}{D^C}\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathit{P}\left(\frac{N^C}{D^C}\right)\mathbf{=}\mathbf{0}$

Short answer

The desired probability is 0.0063.

Step-by-step solution

Important formula

The Baye’s formula is

$\begin{array}{rcl}\mathbf{P}\left(\frac{{\mathbf{B}}_{\mathbf{i}}}{\mathbf{A}}\right)& \mathbf{=}& \frac{\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{i}}\cap \mathbf{A}\mathbf{)}}{\mathbf{P}\mathbf{(}\mathbf{A})}\\ & \mathbf{=}& \frac{\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{i}}\mathbf{)}\mathbf{P}\left(\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{i}}}\right)}{\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{1}}\mathbf{)}\mathbf{P}\left(\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{1}}}\right)\mathbf{+}\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{2}}\mathbf{)}\mathbf{P}\left(\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{2}}}\right)\mathbf{+}...\mathbf{+}\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{k}}\mathbf{)}\mathbf{P}\left(\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{k}}}\right)}\\ & & \end{array}$

Find the probability 

The given values are

$P\left(D\right)=\frac{1}{180},P\left(D^C\right)=\frac{79}{80},P\left(\frac{N}{D}\right)=\frac{1}{2},P\left(\frac{N^C}{D}\right)=\frac{1}{2},P\left(\frac{N}{D^C}\right)=1,P\left(\frac{N^C}{D^C}\right)=0$