2.1.21 problem 28

(a)

$\frac{P\left(D\right|T)}{P\left(D^c\right|T)}=\frac{P\left(D\right)}{P\left(D^c\right)}\frac{P\left(T\right|D)}{P\left(T^c\right|D^c)}.$

(b)

Suppose our population consists of $10000$ people, and only one percent of them is afflicted with the disease. So, $100$ people have the disease and $9900$ people don’t. Suppose the specificity and sensitivity of our test are $95$ percent. Then, out of the $100$ people who have the disease, $95$ test positive and $5$ test negative, and out of the $9900$ people who do not have the disease, $9405$ test negative and $495$ test positive.

Thus, $P\left(D\right|T)=\frac{95}{95+495}.$

Here, we can see why specificity matters more than sensitivity. Since, the disease is rare, most people do not have it. Since specificity is measured as a percentage of the population that doesn’t have the disease, small changes in specificity equate to much larger changes in the number of people than in the case of sensitivity.